BERKELEY 

LIBRARY 

UNIVERSITY  OF 
CALIFORNIA 


ASTRONOMY 
LIBRARY 


<gtft  Of 


Club 


CLASS. 
ACC 


GIFT  OF 
BOHEMIAN  CLUB 


FRONTISPIECE. 
THE  GREAT  TELESCOPE  OF  THE  LICK  OBSERVATORY,  MT.  HAMILTON,  CAL. 

Object-Glass  made  by  A.  Clark  &  Sons  :  Aperture,  36  in.;  Focal  Length,  56  ft.  2  in. 
Mounting  by  Warner  &  Swasey. 


A  TEXT-BOOK 


OF 


ASTRONOMY 


FOR 


COLLEGES  AND  SCIENTIFIC  SCHOOLS 


BY 


CHAKLES   A.  YOUNG,  PH.D.,  LL.D. 

PROFESSOR    OF    ASTRONOMY    IN    PRINCETON    UNIVERSITY 


REVISED   EDITION 


GINN  &  COMPANY 

BOSTON  •  NEW  YORK  •  CHICAGO  •  LONDON 


ASTRONOMY  DIPT, 


ENTERED  AT  STATIONERS'  HALL 


COPYRIGHT,  1888,  BY 
CHARLES    A.   YOUNG 

COPYRIGHT,  1898,  BY 
CHARLES    A.   YOUNG 


ALL  RIGHTS  RESERVED 
613.11 


TEfte  gtftcnaum 

GINN   &   COMPANY  .  PRO- 
PRIETORS  •  BOSTON  •  U.S.A. 


ASTRONO/AY 


PREFACE  TO  FIRST  EDITION. 


THE  present  work  is  designed  as  a  text-book  of  Astronomy 
suited  to  the  general  course  in  our  colleges  and  schools  of 
science,  and  is  meant  to  supply  that  amount  of  information 
upon  the  subject  which  may  fairly  be  expected  of  every 
"  liberally  educated "  person.  While  it  assumes  the  previ- 
ous discipline  and  mental  maturity  usually  corresponding  to 
the  latter  years  of  the  college  course,  it  does  not  demand  the 
peculiar  mathematical  training  and  aptitude  necessary  as  the 
basis  of  a  special  course  in  the  science  —  only  the  most  ele- 
mentary knowledge  of  Algebra,  Geometry,  and  Trigonometry 
is  required  for  its  reading.  Its  aim  is  to  give  a  clear,  accu- 
rate, and  justly  proportioned  presentation  of  astronomical 
facts,  principles,  and  methods  in  such  a  form  that  they  can 
be  easily  apprehended  by  the  average  college  student  with  a 
reasonable  amount  of  effort. 

The  limitations  of  time  are  such  in  our  college  course  that 
probably  it  will  not  be  possible  in  most  cases  for  a  class  to 
take  thoroughly  everything  in  the  book.  The  fine  print  is  to 
be  regarded  rather  as  collateral  reading,  important  to  a  com- 
plete view  of  the  subject,  but  not  essential  to  the  course. 
Some  of  the  chapters  can  even  be  omitted  in  cases  where  it 
is  found  necessary  to  abridge  the  course  as  much  as  possible ; 
e.g.,  the  chapters  on  Instruments  and  on  Perturbations. 

While  the  work  is  no  mere  compilation,  it  makes  no  claims 
to  special  originality :  information  and  help  have  been  drawn 
from  all  available  sources.  The  author  is  under  great  obliga- 
tions to  the  astronomical  histories  of  Grant  and  Wolf,  and 
especially  to  Miss  Clerke's  admirable  "  History  of  Astronomy 
in  the  Nineteenth  Century."  Many  data  also  have  been  drawn 
from  Houzeau's  valuable  "  Vade  Mecum  de  1'Astronome." 


iy  PKEFACE. 

It  has  been  intended  to  bring  the  book  well  down  to  date, 
and  to  indicate  to  the  student  the  sources  of  information  on 
subjects  which  are  necessarily  here  treated  inadequately  on 
account  of  the  limitations  of  time  and  space. 

Special  acknowledgments  are  due  to  Professor  Langley  and 
to  his  publishers,  Messrs.  Ticknor  &  Co.,  for  the  use  of  a 
number  of  illustrations  from  his  beautiful  book,  "The  New 
Astronomy  " ;  and  also  to  D.  Appleton  &  Co.  for  the  use  of 
several  cuts  from  the  author's  little  book  on  the  Sun.  Pro- 
fessor Trowbridge  of  Cambridge  kindly  provided  the  original 
negative  from  which  was  made  the  cut  illustrating  the  com- 
parison of  the  spectrum  of  iron  with  that  of  the  sun.  Warner 
&  Swasey  of  Cleveland  and  Fauth  &  Co.  of  Washington  have 
also  furnished  the  engravings  of  a  number  of  astronomical 
instruments. 

Professors  Todd,  Emerson,  Upton,  and  McNeill  have  given 
most  valuable  assistance  and  suggestions  in  the  revision  of  the 
proof ;  as  indeed,  in  hardly  a  less  degree,  have  several  others. 

PRINCETON,  N.  J.,  August,  1888. 


PREFACE  TO  THE  REVISED  EDITION. 


THE  progress  of  Astronomy  has  been  very  rapid  since  the 
first  publication  of  this  book  in  1889,  and,  although  in  the 
meantime  the  author  has  attempted  as  far  as  possible  to  keep 
the  successive  issues  "  up  to  date  "  by  minor  changes,  notes, 
and  "  addenda,"  it  has  at  last  become  'imperative  to  give 
the  work  a  thorough  revision,  rewriting  certain  portions  and 
making  considerable  additions,  in  order  to  embody  the  new 
and  important  results  which  have  been  obtained  during  the 
last  ten  years. 

The  Appendix  has  also  been  enlarged  by  several  articles 
giving  the  demonstration  of  certain  fundamental  methods  and 


PREFACE.  V 

formulae  for  which,  in  previous  editions,  the  student  was 
referred  to  other  works  not  always  conveniently  accessible. 
In  one  or  two  of  these  articles  the  Calculus  is  necessarily 
used. 

The  various  tables  have  been  corrected  to  correspond  with 
the  latest  and  most  authoritative  data ;  and  a  set  of  illustrative 
exercises  has  been  added  at  the  end  of  nearly  every  chapter. 

While  the  book  has  thus  been  necessarily  somewhat  increased 
in  size,  the  changes  have  been  so  managed  that  no  serious  diffi- 
culty will  be  encountered  in  using  the  new  edition  along  with 
the  older  issues.  The  original  numbering  of  the  articles  has 
been  retained  throughout,  with  only  one  or  two  exceptions, 
the  interpolated  matter  being  designated  by  numbers  with 
asterisks. 

It  is  believed  that  the  book,  so  far  as  its  scope  extends, 
may  now  be  taken  as  fairly  representing  the  present  state  of 
the  science,  although  some  of  the  most  important  recent  dis- 
coveries are  hardly  made  so  prominent  as  would  have  been  the 
case  if  the  revision  had  not  been  substantially  completed  and 
prepared  for  the  press  more  than  two  years  ago;  the  actual 
printing  having  been  much  delayed  by  various  causes. 

Special  acknowledgments  are  due  from  the  author  to  the 
publishers  for  the  liberality  with  which  they  have  made  the 
extensive  and  expensive  changes  in  the  plates,  and  to  Apple- 
ton  &  Co.,  and  Professors  Frost,  Hale,  Holden,  and  Pickering 
for  many  of  the  new  illustrations. 

PRINCETON  UNIVERSITY,  March,  1898. 


PREFACE   TO   ISSUE   OF   1904. 

IN  this  issue  of  the  Revised  Edition  a  considerable  number 
of  corrections,  changes,  and  additions  have  been  made  in  the 
text,  and  three  Addenda  have  been  appended,  in  order  to  bring 
the  book  up  to  date  as  far  as  possible. 

AUGUST,  1904. 


TABLE  OF  CONTENTS. 


PAGES 

INTRODUCTION     .  1-4 

CHAPTER  I.  —  THE  DOCTRINE  OF  THE  SPHERE  :  Definitions  and  Gen- 
eral Considerations 5-20 

CHAPTER  II.  —  ASTRONOMICAL  INSTRUMENTS  :  the  Telescope ;  Time- 
keepers and  Chronograph ;  the  Transit  Instrument  and  Accessories ; 
the  Meridian  Circle  and  Reading  Microscope ;  the  Altitude  and  Azi- 
muth Instrument ;  the  Equatorial  Instrument  and  Micrometer ;  the 
Sextant 21-57 

CHAPTER  III. — CORRECTIONS  TO  ASTRONOMICAL  OBSERVATIONS:  the 
Dip  of  the  Horizon ;  Parallax ;  Semi-diameter ;  Refraction ;  Twilight ; 
Exercises  on  Chapters  I,  II,  and  III  .  .  .  .  .  .  .  58-71 

CHAPTER  IV.  —  PROBLEMS  OF  PRACTICAL  ASTRONOMY  :  the  Determi- 
nation of  Latitude  and  its  Variation,  of  Time,  of  Longitude,  of  a 
Ship's  Place  at  Sea,  of  Azimuth,  and  of  the  Apparent  Right  Ascen- 
sion and  Declination  of  a  Heavenly  Body ;  the  Time  of  Sunrise  or 
Sunset ;  the  Rising  and  Setting  of  a  Star  or  of  the  Moon;  Exercises  .  72-96 

CHAPTER  V.— THE  EARTH:  the  Approximate  Determination  of  its 
Dimensions  and  Form ;  Proofs  of  its  Rotation  ;  Accurate  Determina- 
tion of  its  Dimensions  by  Geodetic  Surveys  and  Pendulum  Observa- 
tions ;  Determination  of  its  Mass  and  Density ;  Exercises  .  97-124 

CHAPTER  VI. —THE  EARTH'S  ORBITAL  MOTION:  the  Motion  of  the 
Sun  among  the  Stars ;  the  Equation  of  Time ;  Precession ;  Nutation  ; 
Aberration  ;  the  Calendar ;  Exercises 125-154 

CHAPTER  VII. —THE  MOON:  her  Orbital  Motion;  Distance  and  Di- 
mensions; Mass,  Density,  and  Superficial  Gravity;  Rotation  and 
Librations ;  Phases ;  Light  and  Heat ;  Physical  Condition ;  Influence 
exerted  on  the  Earth ;  Surface  Structure ;  Possible  Changes  ; 
Exercises  .  .  .  .  .  .  .  .  ...  155-183 

CHAPTER  VIII.  —THE  SUN  :  Distance  and  Dimensions ;  Mass  and  Den- 
sity ;  Rotation ;  Solar  Eye-pieces,  and  Study  of  the  Sun's  Surface ; 
General  Views  as  to  Constitution ;  Sun  Spots,  their  Appearance, 
Nature,  Distribution,  and  Periodicity  ;  the  Spectroscope  .  184-208 


Vlll  TABLE    OF    CONTENTS. 

CHAPTER  IX.  — THE  SPECTROSCOPE  AND  THE  SOLAR  SPECTRUM  :  Chemi- 
cal Elements  present  in  the  Sun  ;  the  Sun-spot  Spectrum  ;  Doppler's 
Principle  ;  the  Chromosphere  and  Prominences ;  the  Corona ;  Exer- 
cises on  Chapters  VIII.  and  IX 209-231 

CHAPTER  X.  —THE  SUN'S  LIGHT  AND  HEAT  :  Comparison  of  Sunlight 
with  Artificial  Lights;  the  Measurement  of  the  Sun's  Heat  and  Deter- 
mination of  the  Solar  Constant ;  the  Pyrheliometer,  Actinometer,  and 
Bolometer ;  the  Sun's  Temperature  ;  Maintenance  of  the  Sun's  Radia- 
tion ;  Conclusions  as  to  its  Age  and  Future  Endurance  .  .  232-247 

CHAPTER  XL  —  ECLIPSES  :  Form  and  Dimensions  of  Shadows ;  Lu- 
nar Eclipses ;  Solar  Eclipses,  Total,  Annular,  and  Partial ;  Ecliptic 
Limits,  and  Number  of  Eclipses  in  a  Year  ;  the  Saros  ;  Occupations  ; 
Exercises t  .  .  .  248-268 

CHAPTER  XII.  —  CENTRAL  FORCES  :  Equable  Description  of  Areas ; 
Areal,  Linear,  and  Angular  Velocities ;  Kepler's  Laws  and  Infer- 
ences from  them  ;  Gravitation  demonstrated  by  the  Moon's  Motion ; 
Conic  Sections  as  Orbits  ;  the  Problem  of  Tw'o  Bodies ;  the  "  Velocity 
from  Infinity"  and  its  Relation  to  the  Species  of  Orbit  described 
by  a  Body  moving  under  Gravitation ;  Intensity  of  Gravitation ; 
Exercises 269-293 

CHAPTER  XIII.  —THE  PROBLEM  OF  THREE  BODIES  :  Disturbing  Forces ; 

Lunar  Perturbations  and  the  Tides 294-316 

CHAPTER  XIV.  —  THE  PLANETS  :  their  Motions,  Apparent  and  Real : 
the  Ptolemaic,  Tychonic,  and  Copernican  Systems ;  the  Orbits  and 
their  Elements ;  Planetary  Perturbations ;  Exercises  .  .  317-339 

CHAPTER  XV.  —  THE  PLANETS  THEMSELVES  :  Methods  of  determining 
their  Diameters,  Masses,  Densities,  Times  of  Rotation,  etc.  ;  the 
"  Terrestrial  Planets,"  —  Mercury,  Venus,  and  Mars ;  the  Asteroids ; 
Intra-Mercurial  Planets  and  the  Zodiacal  Light ;  Exercises  .  340-377 

CHAPTER  XVI.  —THE  MAJOR  PLANETS  :  Jupiter,  Saturn,  Uranus,  and 

Neptune ;  Exercises 378-406 

CHAPTER  XVII.— THE  DETERMINATION  OF  THE  SUN'S  HORIZONTAL 
PARALLAX  AND  DISTANCE  :  Oppositions  of  Mars  and  Transits  of 
Venus ;  Gravitational  Methods ;  Determination  by  Means  of  the 
Velocity  of  Light ;  Exercises 407-427 

CHAPTER  XVIII.  —  COMETS  :  their  Number,  Motions,  and  Orbits ; 
their  Constituent  Parts  and  Appearance;  their  Spectra,  Physical 
Constitution,  and  Probable  Origin ;  Exercises  .  .  .  428-464 

CHAPTER  XIX.  —  METEORS  :  Aerolites,  their  Fall  and  Physical  Char- 
acteristics; Shooting  Stars  and  Meteoric  Showers;  Connection  be- 
tween Meteors  and  Comets ;  Exercises  .....  465-482 


TABLE    OF    CONTENTS.  IX 

CHAPTER  XX.  —  THE  STARS  :  their  Nature  and  Number  ;  the  Constel- 
lations; Star-catalogues;  Stellar  Photography ;  Designation  of  Stars; 
their  Proper  Motions ;  Radial  Motion,  or  Motion  in  Line  of  Sight ; 
the  Motion  of  the  Sun  in  Space  ;  Stellar  Parallax  ;  Exercises  .  483-506 

CHAPTER  XXI.  —  THE  LIGHT  OF  THE  STARS  :  Star  Magnitudes  and 
Photometry  ;  Variable  Stars ;  Stellar  Spectra ;  Scintillation  of  the 
Stars ;  Exercises  .„.<,.....  507-540 

CHAPTER  XXII.  —  AGGREGATIONS  OF  STARS:  Double  and  Multiple 
Stars  ;  Clusters ;  Nebulae ;  Photography  of  Nebulas  ;  the  Milky  Way, 
and  Distribution  of  Stars  in  Space  ;  Constitution  of  the  Stellar  Uni- 
verse ;  Cosmogony  and  the  Nebular  Hypothesis  .  .  .  541-578 

APPENDIX.  —  Reduction  of  Sidereal  Time  to  Solar  ;  Azimuthal  Motion 
of  Star  at  the  Horizon ;  Kepler's  Problem  and  its  Solution,  Numer- 
ically and  by  the  Curve  of  Sines;  Projection  and  Calculation  of 
Lunar  Eclipses ;  Proof  that  the  Orbit  of  a  Body  moving  under  the 
Law  of  Gravitation  is  a  Focal  Conic ;  Expression  for  Velocity  at  any 
Point  of  Orbit ;  Apparent  Epicycloidal  Motion  of  Planets  .  581-599 

ASTRONOMICAL  CONSTANTS,  TABLES          .         .  (      .         .         601-611 

INDEX 613-627 

SUPPLEMENTARY  INDEX  628-630 


INTRODUCTION. 


1.  ASTRONOMY  (ao-rpov  vo'/xo?)  is  the  science  which  treats  of  the 
heavenly  bodies.  As  such  bodies  we  reckon  the  sun  and  moon,  the 
planets  (of  which  the  earth  is  one)  and  their  satellites,  comets  and 
meteors,  and  finally  the  stars  and  nebulae. 

We  have  to  consider  in  Astronomy  :  — 

(a)  The  motions  of  these  bodies,  both  real  and  apparent,  and  the 
laws  which  govern  these  motions. 

(6)   Their  forms,  dimensions,  and  masses. 

(c)  Their  nature,  constitution,  and  conditions. 

(d)  The  effects  they  produce  upon  each  other  by  their  attractions, 
radiations,  or  by. any  other  ascertainable  influence. 

It  was  an  early,  and  has  been  a  most  persistent,  belief  that  the 
heavenly  bodies  have  a  powerful  influence  upon  human  affairs,  so 
that  from  a  knowledge  of  their  positions  and  "aspects"  at  critical 
moments  (as  for  instance  at  the  time  of  a  person's  birth)  one  could 
draw  up  a  "horoscope"  which  would  indicate  the  probable  future. 

The  pseudo-science  which  was  founded  on  this  belief  was  named 
Astrology,  —  the  elder  sister  of  Alchemy,  —  and  for  centuries  As- 
tronomy was  its  handmaid;  i.e.,  astronomical  observations  and  cal- 
culations were  made  mainly  in  order  to  supply  astrological  data. 

At  present  the  end  and  object  of  astronomical  study  is  chiefly 
knowledge  pure  and  simple";  so  far  as  now  appears,  its  development 
has  less  direct  bearing  upqn  the  material  interests  of  mankind  than 
that  of  any  other  of  the  natural  sciences.  It  is  not  likely  that  great 
inventions  and  new  arts  will  grow  out  of  its  laws  and  principles,  such 
as  are  continually  arising  from  physical,  chemical,  and  biological 
discoveries,  though  of  course  it  would  be  rash  to  say  that  such  out- 
growths are  impossible.  But  the  student  of  Astronomy  must  expect 
his  chief  profit  to  be  intellectual,  in  the  widening  of  the  range  of 
thought  and  conception,  in  the  pleasure  attending  the  discovery  of 
simple  law  working  out  the  most  complicated  results,  in  the  delight 


2  INTRODUCTION. 

over  the  beauty  and  order  revealed  by  the  telescope  in  systems  other- 
wise invisible,  in  the  recognition  of  the  essential  unity  of  the  material 
universe,  and  of  the  kinship  between  his  own  mind  and  the  infinite 
Reason  that  formed  all  things  and  is  immanent  in  them. 

At  the  same  time  it  should  be  said  at  once  that,  even  from  the 
lowest  point  of  view,  Astronomy  is  far  from  a  useless  science.  The 
art  of  navigation  depends  for  its  very  possibility  upon  astronomical 
prediction.  Take  away  from  mankind  their  almanacs,  sextants,  and 
chronometers,  and  commerce  by  sea  would  practically  stop.  The 
science  also  has  important  applications  in  the  survey  of  extended 
regions  of  country,  and  the  establishment  of  boundaries,  to  say 
nothing  of  the  accurate  determination  of  time  and  the  arrangement 
of  the  calendar. 

It  need  hardly  be  said  that  Astronomy  is  not  separated  from  kin- 
dred sciences  by  sharp  boundaries.  It  would  be  impossible,  for  in- 
stance, to  draw  a  line  between  Astronomy  on  one  side  and  Geology 
and  Physical  Geography  on  the  other.  Many  problems  relating  to 
the  formation  and  constitution  of  the  earth  belong  alike  to  all  three. 

2.  Astronomy  is  divided  into  many  branches,  some  of  which,  as 
ordinarily  recognized,  are  the  following :  — 

1.  Descriptive  Astronomy,  —  This,  as  its  name  implies,  is  merely 
an  orderly  statement  of  astronomical  facts  and  principles. 

2.  Practical  Astronomy.  —  This  is  quite   as   much   an    art   as   a 
science,  and  treats  of  the  instruments,  the  methods  of  observation, 
and  the  processes  of  calculation  by  which  astronomical  facts  are 
ascertained. 

3.  Theoretical  Astronomy,  which  treats  of  the  calculations  of  orbits 
and  ephemerides,  including  the  effects  of  so-called  "  perturbations." 

4.  Mechanical  Astronomy,  which  is  simply  the  application  of  me- 
chanical principles  to  explain  astronomical  facts  (chiefly  the  planetary 
and  lunar  motions) .    It  is  sometimes  called  Gravitational  Astronomy, 
because,  with  few  exceptions,  gravitation  is  the  only  force  sensibly 
concerned  in  the  motions  of  the  heavenly  bodies.     Until  within  thirty 
years  this  branch  of  the  science  was  generally  designated  as  Physical 
Astronomy,  but  the  term  is  now  objectionable  because  of  late  it  has 
been  used  by  many  writers  to  denote  a  very  different  and  compara- 
tively new  branch  of  the  science  ;  viz., — 


INTRODUCTION.  3 

5.  Astronomical  Physics,  or  Astro-physics.  —  This  treats  of  the 
physical  characteristics  of  the  heavenly  bodies,  their  brightness  and 
spectroscopic   peculiarities,   their   temperature   and   radiation,   the 
nature  and  condition  of  their  atmospheres  and  surfaces,  and  all 
phenomena  which  indicate  or  depend  on  their  physical  condition. 

6.  Spherical  Astronomy.  —  This,  discarding  all  consideration  of 
absolute    dimensions    and    distances,   treats    the   heavenly   bodies 
simply  as  objects  moving  on  the  "  surface  of  the  celestial  sphere  "  : 
it  has  to  do  only  with  angles  and  directions,  and,  strictly  regarded, 
is  in  fact  merely  Spherical  Trigonometry  applied  to  Astronomy. 

3.  The  above-named  branches  are  not  distinct  and  separate,  but 
they  overlap  in  all  directions.  Spherical  Astronomy,  for  instance, 
finds  the  demonstration  of  many  of  its  formulae  in  Gravitational 
Astronomy,  and  their  application  appears  in  Theoretical  and  Prac- 
tical Astronomy.  But  valuable  works  exist  bearing  all  the  differ- 
ent titles  indicated  above,  and  it  is  important  for  the  student  to 
know  what  subjects  he  may  expect  to  find  discussed  in  each ;  for 
this  reason  it  has  seemed  worth  while  to  name  and  define  the 
several  branches,  although  they  do  not  distribute  the  science  be- 
tween them  in  any  strictly  logical  and  mutually  exclusive  manner. 

In  the  present  text-book  little  regard  will  be  paid  to  these  sub- 
divisions, since  the  object  of  the  work  is  not  to  present  a  complete 
and  profound  discussion  of  the  subject  such  as  would  be  demanded 
by  a  professional  astronomer,  but  only  to  give  so  much  knowledge 
of  the  facts  and  such  an  understanding  of  the  principles  of  the 
science  as  may  fairly  claim  to  be  elements  in  a  liberal  education. 
If  this  result  is  gained  in  the  reader's  case,  it  may  easily  happen 
that  he  will  wish  for  more  than  he  can  find  in  these  pages,  and 
then  he  must  have  recourse  to  works  of  a  higher  order  and  far 
more  difficult,,  dealing  with  the  subject  more  in  detail  and  more 
thoroughly. 

To  master  the  present  book  no  further  preparation  is  necessary 
than  a  very  elementary  knowledge  of  Algebra,  Geometry,  and 
Trigonometry,  and  a  similar  acquaintance  with  Mechanics  and 
Physics,  especially  Optics.  While  nothing  short  of  high  mathe- 
matical attainments  will  enable  one  to  become  eminent  in  the  sci- 
ence, yet  a  perfect  comprehension  of  all  its  fundamental  methods 
and  principles,  and  a  very  satisfactory  acquaintance  with  its  main 
results,  is  quite  within  the  reach  of  every  person  of  ordinary  intel- 
ligence, without  any  more  extensive  training  than  may  be  had  in 


4  INTRODUCTION. 

our  common  schools.  At  the  same  time  the  necessary  statements 
and  demonstrations  are  so  much  facilitated  by  the  use  of  trigono- 
metrical terms  and  processes  that  it  would  be  unwise  to  dispense 
with  them  entirely  in  a  work  to  be  used  by  pupils  who  have  already 
become  acquainted  with  them. 

In  discussing  the  different  subjects  which  present  themselves, 
the  writer  will  adopt  whatever  plan  appears  best  fitted  to  convey 
to  the  student  clear  and  definite  ideas,  and  to  impress  them  upon 
the  mind.  Usually  it  will  be  best  to  proceed  in  the  Euclidean 
order,  by  first  stating  the  fact  or  principle  in  question,  and  then 
explaining  its  demonstration.  But  in  some  cases  the  inverse  pro- 
cess is  preferable,  and  the  conclusion  to  be  reached  will  appear 
gradually  unfolding  itself  as  the  result  of  the  observations  upon 
which  it  depends,  just  as  its  discovery  came  about. 


The  occasional  references  to  "  Physics  "  refer  to  the  "  Elementary 
Text-Book  of  Physics,"  by  Anthony  and  Brackett ;  Magie's  revised 
edition,  1897.  John  Wiley  &  Sons,  N.Y. 


THE    "  DOCTRINE   OF    THE    SPHERE. 


CHAPTER  I. 

THE  "DOCTRINE  OF  THE  SPHERE,"  DEFINITIONS,  AND  GENERA: 
CONSIDERATIONS. 

ASTRONOMY,  like  all  the  other  sciences,  has  a  terminology  of  its 
own,  and  uses  technical  terms  in  the  description  of  its  facts  and 
phenomena.  In  a  popular  essay  it  would  of  course  be  proper  to 
avoid  such  terms  as  far  as  possible,  even  at  the  expense  of  circum- 
locutions and  occasional  ambiguity ;  but  in  a  text-book  it  is  desirable 
that  the  reader  should  be  introduced  to  the  most  important  of  them 
at  the  very  outset,  and  made  sufficiently  familiar  with  them  to  use 
them  intelligently  and  accurately. 

4.  The  Celestial  Sphere. — To  ar<  observer  looking  up  to  the 
heavens  at  night  it  seems  as  if  the  stars  were  glittering  points 
attached  to  the  inner  surface  of  a  dome  ;  since  we  have  no  direct  per- 
ception of  their  distance  there  is  no  reason  to  imagine  some  nearer 
than  others,  and  so  we  involuntarily  think  of  the  surface  as  spherical 
with  ourselves  in  its  centre.  Or  if  we  sometimes  feel  that  the  stars 
and  other  objects  in  the  sky  really  differ  in  distance,  we  still  instinc- 
tively imagine  an  immense  sphere  surrounding  and  enclosing  all. 
Upon  this  sphere  we  imagine  lines  and  circles  traced,  resembling 
more  or  less  the  meridians  and  parallels  upon  the  surface  of  the  earth, 
and  by  reference  to  these  circles  we  are  able  to  describe  intelligently 
the  apparent  positions  and  motions  of  the  heavenly  bodies. 

This  celestial  sphere  may  be  regarded  in  either  of  two  different 
ways,  both  of  which  are  correct  and  lead  to  identical  results. 

(a)  We  may  imagine  it,  in  the  first  place,  as  transparent,  and  of 
merely  finite  (though  undetermined)  dimensions,  but  in  some  way 
so  attached  to,  and  connected  with,  the  observer  that  his  eye  always 
remains  at  its  centre  wherever  he  goes.     Each  observer,  in  this  way 
of  viewing  it,  carries  his  own  sky  with  him,  and  is  the  centre  of  his 
own  heavens. 

(b)  Or,  in  the  second  place,  —  and  this  is  generally  the  more  con- 
venient way  of  regarding  the  matter, —  we  may  consider  the  celestial 


6  THE    "  DOCTRINE    OF    THE    SPHERE." 

sphere  as  mathematically  infinite  in  its  dimensions  :  then,  let  the 
observer  go  where  he  will,  he  cannot  sensibly  get  away  from  its 
centre.  Its  radius  being  "  greater  than  any  assignable  quantity," 
the  size  of  continents,  the  diameter  of  the  earth,  the  distance  of  the 
sun,  the  orbits  of  planets  and  comets,  even  the  spaces  between  the 
stars,  are  all  insignificant,  and  the  whole  visible  universe  shrinks 
relatively  to  a  mere  point  at  its  centre.  In  what  follows  we  shall 
use  this  conception  of  the  celestial  sphere.1 

The  apparent  place  of  any  celestial  body  will  then  be  the  point 
on  the  celestial  sphere  where  the  line  drawn  from  the  eye  of  the 
observer  in  the  direction  in  which  he  sees  the  object,  and  produced 
indefinitely,  pierces  the  sphere.  Thus,  in  Figure  1,  A,  B,  C  are 

the  apparent  places  of  a,  b,  and  c, 
the  observer  being  at  0.  The  ap- 
parent place  of  a  heavenly  body 
evidently  depends  solely  upon  its 
direction,  and  is  wholly  independent 
of  its  distance  from  the  observer. 

5.  Linear  and  Angular  Dimensions. 

—  Linear  dimensions  are  such  as  may 
be  expressed  in  linear  units ;  i.e.,  in 
miles,  feet,  or  inches;  in  metres  or 

millimetres.       Angular    dimensions 
FIG.  i.  ,  . 

are  expressed  in  angular  units ;  i.e., 

in  right  angles,  in  radians,2  or  (more  commonly  in  astronomy)  in 
degrees,  minutes,  and  seconds.  Thus,  for  instance,  the  linear  semi- 


1  To  most  persons  the  sky  appears,  not  a  true  hemisphere,  but  a  flattened 
vault,  as  if  the  horizon  were  more  remote  than  the  zenith.     This  is  a  subjective 
effect  due  mainly  to  the  intervening  objects  between  us  and  the  horizon.     The 
sun  and  moon  when  rising  or  setting  look  much  larger  than  when  they  are 
higher  up,  for  the  same  reason. 

2  A  radian  is  the  angle  which  is  measured  by  an  arc  equal  in  length  to  radius. 
Since  a  circle  whose  radius  is  unity  has  a  circumference  of  2  ?r,  and  contains  360°, 

or  21,600/,  or  1,296,000",  it  follows  that  a  radian  contains  (^:)  i  °r  (   27r   )  ' 


or  .  ^  (approximately)?  a  radian  =  57>3o  _  3437.7-  =  206264.8". 

Hence,  to  reduce  to  seconds  of  arc  an  angle  expressed  in 
radians,  we  must  multiply  it  by  the  number  206264.8;  a 
relation  of  which  we  shall  have  to  make  frequent  use. 


THE 

diameter  of  the  sun  is  about  697,000  kilometres  (433,000  miles), 
while  its  angular  semidiameter  is  about  16',  or  a  little  more  than 
a  quarter  of  a  degree.  Obviously,  angular  units  alone  can  properly 
be  used  in  describing  apparent  distances  and  dimensions  in  the  sky. 
For  instance,  one  cannot  say  correctly  that  the  two  stars  which  are 
known  as  "  the  pointers  "  are  two  or  five  or  ten  feet  apart :  their 
distance  is  about  five  degrees. 

It  is  sometimes  convenient  to  speak  of  "  angular  area,"  the  unit 
of  which  is  a  "  square  degree  "  or  a  "  square  minute  "  ;  i.e.,  a  small 
square  in  the  sky  of  which  each  side  is  1°  or  1'.  Thus  we  may 
compare  the  angular  area  of  the  constellation  Orion  with  that  of 
Taurus,  in  square  degrees,  just  as  we  might  compare  Pennsylvania 
and  New  Jersey  in  square  miles. 

6.    Relation  between  the  Distance  and  Apparent  Size  of  an  Object. 

—  Suppose  a  globe  having  a  radius  BC  equal  to  r.     As  seen  from 


FIG.  2. 


the  point  A  (Fig.  2)  its  apparent  (i.e.,  angular)  semidiameter  will 
be  BA  C  or  s,  its  distance  being  A  C  or  R. 

We  have  immediately  from  Trigonometry,  since  B  is  a  right  angle, 

R  "o 

If,  as  is  usual  in  Astronomy,  the  diameter  of  the  object  is  small 
as  compared  with  its  distance,  we  may  write  s  =  —  ,  which  gives  s 

in  radians  (not  in  degrees  or  seconds).  If  we  wish  it  in  the  ordi- 
nary angular  units, 

s°  =  57.3^ ,  or  s'  =  3437.7^ ,  or  s"  =  206264.8^ , 
MM  H 

where  s°  means  s  in  degrees  ;  s',  s  in  minutes  ;  s",  s  in  seconds  of  arc. 
In  either  form  of  the  equation  we  see  that  the  apparent  diameter 
varies  directly  as  the  linear  diameter,  and  inversely  as  the  distance. 


8  DEFINITIONS   AND   GENERAL   CONSIDERATIONS. 

In  the  case  of  the  moon,  R  =  about  239,000  miles  ;  and  r,  1081 
miles.  Hence  s  =  ^§f  J^  =  ^T  of  a  radian,  which  is  a  little  more 
than  £  of  a  degree,  or  about  933". 

It  may  be  mentioned  here  as  a  rather  curious  fact  that  most  persons  say 
that  the  moon  appears  about  a  foot  in  diameter ;  at  least,  this  seems  to  be 
the  average  estimate.1  This  implies  that  the  surface  of  the  sky  appears  to 
them  only  about  110  feet  away,  since  that  is  the  distance  at  which  a  disc 
one  foot  in  diameter  would  have  an  angular  diameter  of  T^  of  a  radian, 
or  jo. 

7.  Vanishing  Point.  — Any  system  of  parallel  lines  produced  in 
one  direction  will  appear  to  pierce  the  celestial  sphere  at  a  single 
point.     They  actually  pierce  it  at  different  points,  separated  on  the 
surface  of  the  sphere  by  linear  distances  equal  to  the  actual  dis- 
tances between  the  lines,  but  on  the  infinitely  distant  surface  these 
linear  distances,  being  only  finite,  become  invisible,  subtending  at 
the  centre  angles  less  than  anything  assignable.      The  different 
points,  therefore,  coalesce  into  a  spot  of  apparently  infinitesimal 
size — the  so-called  "vanishing  point'7  of  perspective.     Thus  the 
axis  of  the  earth  and  all  lines  parallel  to  this  axis  point  to  the 
celestial  pole. 

POINTS   AND   CIRCLES   OF   REFERENCE. 

8.  The  Zenith.  —  The  Zenith  is  the  point  vertically  overhead,  i.e., 
the  point  where  a  plumb-line,  produced  upwards,  would  pierce  the 
sky  :  it  is  determined  by  the  direction  of  gravity  where  the  observer 
stands. 

If  the  earth  were  exactly  spherical,  the  zenith  might  also  be  de- 
fined as  the  point  where  a  line  drawn  from  the  centre  of  the  earth 
upward  through  the  observer  meets  the  sky.  But  since,  as  we  shall 
see  hereafter,  the  earth  is  not  an  exact  globe,  this  second  definition 
indicates  a  point  known  as  the  Geocentric  Zenith,  which  is  not  iden- 
tical with  the  True  or  Astronomical  Zenith,  determined  by  the  direc- 
tion of  gravity.  , 

9.  The  Nadir.  —  The  Nadir  is  the  point  opposite  the  zeni th- 
under foot,  of  course. 

Both  zenith  and  nadir  are  derived  from  the  Arabic,  which  lan- 
guage has  also  given  us  many  other  astronomical  terms. 

1  See  note  on  p.  20,  at  the  end  of  the  chapter. 


REFERENCE   POINTS   AND   CIRCLES. 

10.  Horizon.  —  The  Horizon1  is  a  great  circle  of  the  celestial 
sphere,  having  the  zenith  and  nadir  as  its  poles :   it  is  therefore 
half-way  between  them,  and  90°  from  each. 

A  horizontal  plane,  or  the  plane  of  the  horizon,  is  a  plane  perpen- 
dicular to  the  direction  of  gravity,  and  the  horizon  may  also  be 
correctly  denned  as  the  intersection  of  the  celestial  sphere  by  this 
plane. 

Many  writers  make  a  distinction  between  the  sensible  and  rational 
horizons.  The  plane  of  the  sensible  horizon  passes  through  the 
observer  ;  the  plane  of  the  rational  horizon  passes  through  the  cen- 
tre of  the  earth,  parallel  to  the  plane  of  the  sensible  horizon  :  these 
two  planes,  parallel  to  each  other,  and  everywhere  about  4000  miles 
apart,  trace  out  on  the  sky  the  two  horizons,  the  sensible  and  the 
rational.  It  is  evident,  however,  that  on  the  infinitely  distant  sur- 
face of  the  celestial  sphere,  the  two  traces  sensibly  coalesce  into 
one  single  great  circle,  which  is  the  horizon  as  first  defined.  We 
get,  therefore,  but  one  horizon  circle  in  the  sky. 

11.  The  Visible  Horizon  is  the  line  where  sky  and  earth  meet. 
On  land  it  is  an  irregular  line,  broken  by  hills  and  trees,  and  of  no 
astronomical  value  ;  but  at  sea  it  is  a  true  circle,  and  of  great  im- 
portance in  observation.     It  is  not,  however,  a  great  circle,  but, 
technically  speaking,  only  a  small  circle ;  depressed  below  the  true 
horizon   by  an  amount  depending  upon  the  observer's   elevation 
above  the  water.     This  depression  is  called  the  Dip  of  the  Horizon, 
and  will  be  discussed  further  on. 

12.  Vertical  Circles.  —  These  are  great  circles  passing  through 
the  zenith  and  nadir,  and  therefore  necessarily  perpendicular  to  the 
horizon  —  secondaries  to  it,  to  use  the  technical  term. 

Parallels  of  Altitude,  or  Almucantars.  —  These  are  small  circles 
parallel  to  the  horizon  :  the  term  Almucantar  is  seldom  used. 

The  points  and  circles  thus  far  defined  are  determined  entirely 
by  the  direction  of  gravity  at  the  station  occupied  by  the  observer. 


13.  The  Diurnal  Rotation  of  the  Heavens.  —  If  one  watches  the 
sky  for  a  few  hours  some  night,  he  will  find  that,  while  certain  stars 
rise  in  the  east,  others  set  in  the  west,  and  nearly  all  the  constella- 
tions change  their  places.  Watching  longer  and  more  closely,  it  will 

1  Beware  of  the  common,  but  vulgar,  pronunciation,  H6nzon. 


10 


DEFINITIONS   AND    GENERAL   CONSIDERATIONS. 


appear  that  the  stars  move  in  circles,  uniformly,  in  such  a  way  as 
not  to  disturb  their  relative  configurations,  but  as  if  they  were 
attached  to  the  inner  surface  of  a  revolving  sphere,  turning  on  its 
axis  once  a  day.  The  path  thus  daily  described  by  a  star  is  called 
its  " diurnal  circle" 

It  is  soon  evident  that  in  our  latitude  the  visible  "  pole  "  of  this 
sphere  —  the  point  about  which  it  turns  —  is  in  the  north,  not  quite 
half-way  up  from  the  horizon  to  the  zenith,  for  in  that  region  the 
stars  hardly  move  at  all,  but  keep  their  places  all  night  long. 

14,  The  Poles.  —  The  Poles  may  be  defined  as  the  two  points  in 
the  sky,  one  in  the  northern  hemisphere  and  one  in  the  southern, 


FIG.  3.  —  The  Pole  Star  and  the  Pointers. 


where  a  star's  diurnal  circle  reduces  to  zero  ;  i.e.,  points  where,  if  a  star 
were  placed,  it  would  suffer  no  apparent  change  of  place  during  the 
whole  twenty-four  hours.  The  line  joining  these  poles  is,  of  course, 
the  axis  of  the  celestial  sphere,  about  which  it  seems  to  rotate  daily. 

The  exact  place  of  the  pole  may  be  found  by  observing  some  star 
very  near  the  pole  at  two  times  12  hours  apart,  and  taking  the  mid- 
dle point  between  the  two  observed  places  of  the  star. 

The  definition  of  the  pole  just  given  is  independent  of  any  theory 
as  to  the  cause  of  the  apparent  rotation  of  the  heavens.  If,  how- 


REFERENCE   POINTS   AND   CHICLES.  11 

ever,  we  admit  that  it  is  due  to  the  earth's  rotation  on  its  axis,  then 
we  may  define  the  poles  as  the  points  where  the  earth's  axis  produced 
pierces  the  celestial  sphere. 

15.  The  Pole-star  (Polaris). — The  place  of  the  northern  pole  is 
very  conveniently  marked  by  the  Pole-star,  a  star  of  the  second  mag- 
nitude, which  is  now  only  about  1  J°  from  the  pole  :  we  say  now,  be- 
cause on  account  of  a  slow  change  in  the  direction  of  the  earth's 
axis,  called  "precession"  (to  be  discussed  later),  the  distance  be- 
tween the  pole-star  and  the  pole  is  constantly  changing,  and  has  been 
for  several  centuries  gradually  decreasing. 

The  pole-star  stands  comparatively  solitary  in  the  sky,  and  may 
easily  be  recognized  by  means  of  the  so-called  "pointers,"  —  two 
stars  in  the  "  dipper"  (in  the  constellation  of  Ursa  Major)  — which 
point  very  nearly  to  it,  as  shown  in  Fig.  3.  The  pole  is  very  nearly 
on  the  line  joining  Polaris  with  the  star  Mizar  (£  Urs.  Maj.,  at  the 
bend  in  the  handle  of  the  dipper) ,  and  at  a  distance  just  about  one- 
quarter  of  the  distance  between  the  pointers,  which  are  nearly  5° 
apart. 

The  southern  pole,  unfortunately,  is  not  so  marked  by  any  con- 
spicuous star. 

16.  The  Celestial  Equator,  or  Equinoctial  Circle. — This  is  a  great 
circle  midway  between  the  two  poles,  and  of  course  90°  from  each. 
It  may  also  be  defined  as  the  intersection  of  the  plane  of  the  earth's 
equator  with  the  celestial  sphere.     It  derives  its  name  from  the  fact 
that,  at  the  two  dates  in  the  year  when  the  sun  crosses  this  circle  — 
about  March  20  and  Sept.  22 — the  day  and  night  are  equal  in  length. 


17.  The  Vernal  Equinox,  or  First  of  Aries. — The  Equinox,  strictly 
speaking,  is  the  time  when  the  sun  crosses  the  equator,  but  the  term 
has  come  by  accommodation  to  denote  also  the  point  where  it  crosses. 
This  crossing  occurs  twice  a  year,  about  March  20th  and  September 
22d,  and  the  Vernal  Equinox  is  the  point  on  the  equator  where  the 
sun  crosses  it  in  the  spring.  It  is  sometimes  called  the  Greenwich 
of  the  Celestial  Sphere,  because  it  is  used  as  a  reference  point  in  the 
sky,  much  as  Greenwich  is  on  the  earth.  Its  position  is  not  marked 
by  any  conspicuous  star. 

Why  this  point  is  also  called  the  "  First  of  Aries  "  will  appear 
later,  when  we  come  to  speak  of  the  zodiac  and  its  "  signs." 


12  DEFINITIONS   AND   GENERAL   CONSIDERATIONS. 

18.  Hour-Circles. — Hour-circles  are  great  circles  of  the  celestial 
sphere    passing  through  its  poles,   and  consequently   perpendicular 
to  the  celestial  equator.     They  correspond  exactly  to  the  meridians 
of  the  earth,  and  some  writers  call  them  "  Celestial  Meridians"  ;  but 
the   term  is  objectionable,  as  likely  to  lead  to  confusion  with  the 
Meridian,  to  be  noted  immediately. 

19.  The  Meridian  and  Prime  Vertical.  —  The  Meridian  is  the  great 
circle  passing  through  the  pole  and  the  zenith.     Since  it  is  a  great 
circle,  it  must  necessarily  pass  through  both  poles,  and  through  the 
nadir  as  well  as  the  zenith,  and  must  be  perpendicular  both  to  the 
equator  and  to  the  horizon. 

It  may  also  be  correctly  denned  as  the  Vertical  Circle  which  passes 
through  the  pole;  or,  again,  as  the  Hour-Circle  which  passes  through 
the  zenith,  since  all  vertical  circles  must  pass  through  the  zenith,  and 
all  hour-circles  through  the  pole. 

The  Prime  Vertical  is  the  Vertical  Circle  (passing  through  the 
zenith)  at  right  angles  to  the  meridian ;  hence  lying  east  and  west 
on  the  celestial  sphere. 

20.  The  Cardinal  Points. — The  North  and  South  Points  are  the 
points  on  the  horizon  where  it  is  intersected  by  the  meridian ;  the 
East  and  West  Points  are  where  it  is  cut  by  the  prime  vertical,  and 
also  by  the  equator.     The  North  Point,  which  is  on  the  horizon,  must 
not  be  confounded  with  the  North  Pole,  which  is  not  on  the  horizon, 
but  at  an  elevation  equal  (see  Art.  30)  to  the  latitude  of  the  observer. 


"With  these  circles  and  points  of  reference  we  have  now  the  means 
to  describe  intelligibly  the  position  of  a  heavenly  body,  in  several 
different  ways. 

"We  may  give  its  altitude  and  azimuth^  or  its  declination  and  hour- 
angle;  or,  if  we  know  the  time,  its  declination  and  right  ascension. 
Either  of  these  pairs  of  co-ordinates,  as  they  are  called,  will  define 
its  place  in  the  sky. 

21.  Altitude  and  Zenith  Distance  (Fig.  4).  — The  Altitude  of  a 
heavenly  body  is  its  angular  elevation  above  the  horizon,  and  is  meas- 
ured by  the  arc  of  the  vertical  circle  passing  through  the  body,  and 
intercepted  between  it  and  the  horizon. 


CELESTIAL   CO-ORDINATES.  13 

The  Zenith  Distance  of  a  body  is  simply  its  angular  distance  from 
the  zenith,  and  is  the  complement  of  the  altitude.  Altitude  +  Zenith 
Distance  =  90°. 

22.  Azimuth  and  Amplitude  (Fig.  4). — The  Azimuth  of  a  body 
is  the  angle  at  the  zenith,  between  the  meridian  and  the  vertical  circle, 
which  passes  through  the  body.  It  is  measured  also  by  the  arc  of  the 
horizon  intercepted  between  the  north  or  south  point,  and  the  foot 
of  this  vertical.  The  word  is  of  Arabic  origin,  and  has  the  same 
meaning  as  the  true  bearing  in  surveying  and  navigation. 


H 

E 

FIG.  4.  —  The  Horizon  and  Vertical  Circles. 


0,  the  place  of  the  Observer. 
OZ,  the  Observer's  Vertical. 
Z,  the  Zenith;  P,  the  Pole. 
SENW,  the  Horizon. 
SZPN,  the  Meridian. 
EZW,  the  Prime  Vertical. 


M,  some  Star. 

ZMH,  arc  of  the  Star's  Vertical  Circle. 

TMR,  the  Star's  Almucantar. 

Angle  TZM,  or  arc  SWNEH,  Star's  Azimuth. 

Arc  HM,  Star's  Altitude. 

Arc  ZM,  Star's  Zenith  Distance. 


There  are  various  ways  of  reckoning  azimuth.  Many  writers  express  it 
in  the  same  manner  as  the  bearing  is  expressed  in  surveying ;  i.e.,  so  many 
degrees  east  or  west  of  north  or  south ;  1ST.  20°  E.,  etc.  The  more  usual 
way  at  present  is,  however,  to  reckon  it  in  degrees  from  the  south  point  clear 
round  through  the  west  to  the  point  of  beginning :  thus  an  object  in  the 
SW.  would  have  an  azimuth  of  45°;  in  the  NW.,  135°;  in  the  N.,  180°;  in 
the  KE.,  225° ;  and  in  the  SE.,  315°.  For  example,  to  find  a  star  whose 
azimuth  is  260°,  and  altitude  60°,  we  must  face  N.  80°  E.,  and  then  look 
up  two-thirds  of  the  way  to  the  zenith.  The  object  in  this  case  has  an 
amplitude  of  10°  N.  of  E.,  and  a  zenith  distance  of  30°.  Evidently  both 
the  azimuth  and  altitude  of  a  heavenly  body  are  continually  changing. 

The  Amplitude  of  a  body  is  the  angle  intercepted  between  the 
Prime  vertical  and  the  Vertical  circle  which  passes  through  the  body. 


14  DEFINITIONS   AND   GENERAL    CONSIDERATIONS. 

In  Fig.  4,  SENW  represents  the  horizon,  S  being  the  south  point, 
and  Z  the  zenith.  The  angle  SZM,  which  numerically  equals  the 
arc  SH,  is  the  Azimuth  of  the  star  M ;  while  EZM,  or  EH,  is  its 
Amplitude.  MH  is  its  Altitude,  and  ZM  its  Zenith  Distance. 


23.  Declination  and  Polar  Distance  (Fig.  5).  —  The  Declination 
of  a  heavenly  body  is  its  angular  distance  north  or  south  of  the  celes- 
tial equator,  and  is  measured  by  the  arc  of  the  hour-circle  passing 
through  the  object,  intercepted  between  it  and  the  equator.     It  is 
reckoned  positive  (-}-)  north  of  the  celestial  equator,  and  negative  (— ) 
south  of  it.     Evidently  it  is  precisely  analogous  to  the  latitude  of  a 
place  on  the  earth.     The  north-polar  distance  of  a  star  is  its  angular 
distance  from  the  North  Pole,  and  is  simply  the  complement  of  the 
declination.     Declination  -j-  North-Polar  Distance  =  90°. 

The  declination  of  a  star  remains  always  the  same  ;  at  least,  the 
slow  changes  that  it  undergoes  need  not  be  considered  for  our  pres- 
ent purpose.  "  Parallels  of  Declination  "  are  small  circles  parallel 
to  the  celestial  equator. 

24.  The  Hour-Angle  (Fig.  5).  —  The  Hour- Angle  of  a  star  is  the 
angle  at  the  pole  between  the  meridian  and  the  hour-circle  passing 
through  the  star.     It  may  be  reckoned  in  degrees  ;  but  it  also  may 
be,  and  most  commonly  is,  reckoned  in  hours,  minutes,  and  seconds 
of  time  ;    the  hour  being  equivalent  to  fifteen  degrees,  and  the  min- 
ute and  second  of  time  being  equal  to  fifteen  minutes  and  seconds 
of  arc  respectively. 

Of  course  the  hour-angle  of  an  object  is  continually  changing,  be- 
ing zero  when  the  object  is  on  the  meridian,  one  hour,  or  fifteen 
degrees,  when  it  has  moved  that  amount  westward,  and  so  on. 

25.  Right  Ascension  (Fig.  5).  —  The  Eight  Ascension  of  a  star 
is  the  angle  at  the  pole  between  the  star's  hour-circle  and  the  hour- 
circle  (called  the  Equinoctial  Colure),  which  passes  through  the  vernal 
equinox. 

It  may  be  defined  also  as  the  arc  of  the  equator,  intercepted  be- 
tween the  vernal  equinox  and  the  foot  of  the  star's  hour-circle. 

It  is  always  reckoned  from  the  equinox  toward  the  east ;  some- 
times in  degrees,  but  usually  in  hours,  minutes,  and  seconds  of  time. 
The  right  ascension,  like  the  declination,  remains  unchanged  by  the 
diurnal  motion. 


CELESTIAL   CO-ORDINATES. 


15 


26.  .Sidereal  Time  (Fig.  5).  —  For  many  astronomical  purposes 
it  is  convenient  to  reckon  time,  not  by  the  sun's  position  in  the  sky, 
but  by  that  of  the  vernal  equinox. 

The  Sidereal  Time  at  any  moment  may  be  defined  as  the  hour- 
angle  of  the  vernal  equinox.  It  is  sidereal  noon,  when  the  equinoc- 
tial point  is  on  the  meridian  ;  1  o'clock  (sidereal)  when  its  hour- 
angle  is  15° ;  and  23  o'clock  when  its  hour-angle  is  345°,  i.e.,  when 
the  vernal  equinox  is  an  hour  east  of  the  meridian ;  the  time  being 
reckoned  round  through  the  whole  24  hours.  On  account  of  the 
annual  motion  of  the  sun  among  the  stars,  the  Solar  Day,  by  which 


FIG.  5.—  Hour-Circles,  etc. 


O,  place  of  the  Observer ;  Z,  his  Zenith. 

SENW,  the  Horizon. 

POP',  the  Axis  of  the  Celestial  Sphere. 

P  and  P',  the  two  Poles  of  the  Heavens. 

EQWT,  the  Celestial  Equator,  or  Equinoc- 
tial. 

X,  the  Vernal  Equinox,  or  "  First  of  Aries." 

PXP',  the  Equinoctial  Colure,  or  Zero  Hour- 
Circle. 


TO,  some  §tar. 

Ym,  the  Star's  Declination;  Pm,  its  North-- 
polar Distance. 

Angle  mPjR  =  arc  QY,  the  Star's  (eastern) 
Hour-Angle;  =  24h  minus  Star's  (west- 
ern) Hour-Angle. 

Angle  XPm  =  arc  XY,  Star's  Eight  Ascen- 
sion. Sidereal  time  at  the  moment 
=  24h  minus  angle  XPQ. 


time  is  reckoned  for  ordinary  purposes,  is  about^jmniites-Ionger 
thau-tlie^sidereaLday.  The  exact  difference  is  3m568.556  (sidereal), 
or  just  one  day  in  a  year ;  there  being  366J  sidereal  days  in  the 
year,  as  against  365J  solar  days.  See  also  Arts.  110  and  1000. 

27.  Observatory  Definition  of  Right  Ascension.  —  It  is  evident 
from  the  above  definition  of  sidereal  time  that  we  may  also  define  the 
Eight  Ascension  of  a  star  as  the  sidw<Ml__tim&-when  the  star  crosses 
the  meridian.  The  Star  and  the  Vernal  Equinox  are  (practically) 


16  DEFINITIONS    AND   GENERAL   CONSIDERATIONS. 

fixed  points  in  the  sky,  and  do  not  change  their  relative  position 
during  the  sky's  apparent  daily  revolution ;  a  given  star,  therefore, 
always  comes  to  the  meridian  of  any  observer  the  same  number  of 
hours  after  the  vernal  equinox  has  passed  ;  and  this  number  of  hours 
is  the  sidereal  time  at  the  moment  of  the  star's  transit,  and  measures 
its  right  ascension.  In  the  observatory,  this  definition  of  right  as- 
cension is  the  most  natural  and  convenient. 

It  is  obvious  that  the  right  ascension  of  a  star  corresponds  in  the 
sky  exactly  with  the  longitude  of  a  place  on  the  earth ;  terrestrial 
longitude  being  reckoned  from  Greenwich,  just  as  right  ascension  is 
reckoned  from  the  vernal  equinox. 

N.  B.  We  shall  find  hereafter  that  the  heavenly  bodies  have  lati- 
tudes and  longitudes  of  their  own  ;  but  unfortunately  these  celestial 
latitudes  and  longitudes  do  not  correspond  to  the  terrestrial,  and  great 
care  is  necessary  to  prevent  confusion.  (See  Art.  179.) 

28.  An  armillary  sphere,  or  some  equivalent  apparatus,  is  almost 
essential  to  enable  a  beginner  to  get  correct  ideas  of  the  points, 
circles,  and  co-ordinates  defined  above,  but  the  figures  will  perhaps 
be  of  assistance. 

The  first  of  them  (Fig.  4)  represents  the  horizon,  meridian,  and 
prime  vertical,  and  shows  how  the  position  of  a  star  is  indicated  by 
its  altitude  and  azimuth.  This  framework  of  circles,  depending 
upon  the  direction  of  gravity,  to  an  observer  at  any  given  station 
always  remains  apparently  unchanged  in  position,  while  the  sky 
apparently  turns  around  outside  it. 

The  other  figure  (Fig.  5)  represents  the  system  of  points  and 
circles  which  depend  upon  the  earth's  rotation,  and  are  independent 
of  the  direction  of  gravity.  The  vernal  equinox  and  the  hour-circles 
apparently  revolve  with  the  stars  while  the  pole  remains  fixed  upon 
the  meridian,  and  the  equator  and  parallels  of  declination,  revolving 
truly  in  their  own  planes,  also  appear  to  be  at  rest  in  the  sky.  But 
the  whole  system  of  lines  and  points  represented  in  the  figure  (hori- 
zon and  meridian  alone  excepted)  may  be  considered  as  attached  to, 
or  marked  out  upon,  the  inner  surface  of  the  celestial  vault  and 
whirling  with  it. 

It  need  hardly  be  said  that  the  "  appearances  are  deceitful " 
that  which  is  really  carried  around  by  the  earth's  rotation  is  the  ob- 
server, with  his  plumb-line  and  zenith,  his  horizon  and  meridian  ; 
while  the  stars  stand  still  —  at  least,  their  motions  in  a  day  are 
insensible  as  seen  from  the  earth. 


DEFINITIONS   AND   GENERAL   CONSIDERATIONS.  17 

At  the  poles  of  the  earth,  which  are,  mathematically  speaking,  "  singular  " 
points,  the  definitions  of  the  Meridian,  of  North  and  South,  etc.,  break 
down. 

There  the  pole  (celestial)  and  zenith  coincide,  and  any  number  of  circles 
may  be  drawn  through  the  two  points,  which  have  now  become  one.  The 
horizon  and  equator  coalesce,  and  the  only  direction  on  the  earth's  surface 
is  due  south  (or  north)  —  east  and  west  have  vanished. 

A  single  step  of  the  observer  will,  however,  remedy  the  confusion :  zenith 
and  pole  will  separate,  and  his  meridian  will  again  become  determinate. 

29.  To  recapitulate :  The  direction  of  gravity  at  the  point  where 
the  observer  stands  determines  the  Zenith  and  Nadir,  the  Horizon,  and 
the  Almucantars  (parallel  to  the  Horizon),  and  all  the  vertical  circles. 
One  of  the  verticals,  the  Meridian,  is  singled  out  from  the  rest  by 
the  circumstance  that  it  passes  through  the  pole  of  the  sky,  marking 
the  North  and  South  Points  where  it  cuts  the  horizon. 

Altitude  and  Azimuth  (or  their  complements,  Zenith  Distance 
and  Amplitude)  are  the  co-ordinates  which  designate  the  position 
of  a  body  by  reference  to  the  Zenith  and  the  Meridian. 

Similarly,  the  direction  of  the  earth's  axis  (which  is  independent 
of  the  observer's  place  on  the  earth)  determines  the  Poles,  the 
Equator,  the  Parallels  of  Declination,  and  the  Hour-Circles.  Two 
of  these  Hour-Circles  are  singled  out  as  reference  lines ;  one  of  them, 
the  Meridian,  which  passes  through  the  Zenith,  and  is  a  purely 
local  reference  line ;  the  other,  the  Equinoctial  Colure,  which  passes 
through  the  Vernal  Equinox,  a  point  chosen  from  its  relation  to  the 
sun's  annual  motion.  Declination  and  Hour-Angle  are  the  co-ordi- 
nates which  refer  the  place  of  a  star  to  the  Pole  and  the  Meridian ; 
while  Declination  and  Right  Ascension  refer  it  to  the  Pole  and  Equi- 
noctial Colure.  The  latter  are  the  co-ordinates  usually  employed  in 
star-catalogues  and  ephemerides  to  define  the  positions  of  stars  and 
planets,  and  correspond  to  Latitude  and  Longitude  on  the  earth. 


30.  Relation  of  the  Apparent  Diurnal  Motion  of  the  Sky  to  the 
Observer's  Latitude.  —  Evidently  the  apparent  motions  of  the  stars 
will  be  considerably  influenced  by  the  station  of  the  observer,  since 
the  place  of  the  pole  in  the  sky  will  depend  upon  it.  The  Altitude 
of  the  pole,  or  its  height  in  degrees  above  the  horizon,  is  always  equal 
to  the  Latitude  of  the  observer.  Indeed,  the  German  word  for  lati- 
tude (astronomical)  is  Polhohe;  i.e.,  simply  "  Pole-height." 


18 


LATITUDE   AND   THE   POLE. 


This  will  be  clear  from  Fig.  6.  The  latitude  of  a  place  is 
the  angle  between  its  plumb-line  and  the  plane  of  the  equator ;  the 
angle  ONQ  in  the  figure.  [If  the  earth  were  truly  spherical,  N 
would  coincide  with  (7,  the  centre  of  the  earth.  The  ordinary 
definition  of  latitude  given  in  the  geographies  disregards  the  slight 
difference.] 

Now  the  angle  H'OP"  is  equal  to  ONQ,  because  their  sides  are  mu- 
tually perpendicular ;  and  it  is  also  the  altitude  of  the  pole,  because 
the  line  HH'  is  horizontal  at  0,  and  OP",  being  directed  towards  the 
celestial  pole,  is  therefore  parallel  to  pCPP',  the  axis  of  the  earth. 

This  fundamental  relation,  that  the  altitude  of  the  celestial  pole  is 
the  Latitude  of  the  observer,  cannot  be  too  strongbr  impressed  on  the 
student's  mind.  The  usual  symbol  for  the  latitude  of  a  place  is  <j>. 


Fie.  6.  —Relation  of  Latitude  to  the  Elevation  of  the  Pole. 

31.  The  Right  Sphere.  —  If  the  observer  is  situated  at  the 
earth's  equator,  i.e.,  in  latitude  zero  (<£  =  o),  the  pole  will  be  in  the 
horizon,  and  the  equator  will  pass  vertically  overhead  through  the 
zenith. 

The  stars  will  rise  and  set  vertically,  and  their  diurnal  circles  will 
all  be  bisected  by  the  horizon,  so  that  they  will  be  12  hours  above 
it  and  12  below.  This  aspect  of  the  heavens  is  called  the  Right 
Sphere. 


32.  The  Parallel  Sphere.  — If  the  observer  is  at  the  pole  of  the 
earth  (<£  =  90°) ,  then  the  celestial  pole  will  be  in  the  zenith,  and 
the  equator  will  coincide  with  the  horizon.  If  he  is  at  the  North 
Pole,  all  stars  north  of  the  celestial  equator  will  remain  permanently 


DEFINITIONS   AND   GENERAL   CONSIDERATIONS.  19 

above  the  horizon,  never  rising  or  falling  at  all,  but  sailing  around 
on  circles  of  altitude  (or  Almucantars)  parallel  to  the  horizon. 
Stars  in  the  Southern  Hemisphere,  on  the  other  hand,  would  never 
rise  to  view.  As  the  sun  and  the  moon  move  in  such  a  way  that 
during  half  the  time  they  are  alternately  north  and  south  of  the 
equator,  they  will  be  half  the  time  above  the  horizon  and  half  the 
time  below  it.  The  moon  would  be  visible  for  about  a  fortnight  at  a 
time,  and  the  sun  for  six  months. 

33.  The  Oblique  Sphere  (Fig.  7).  —  At  any  station  between  the 
pole  and  equator  the  stars  will  move  in  circles  oblique  to  the  horizon, 
SENW  in  the  figure.  Those  whose  distance  from  the  elevated  pole 
is  less  than  the  latitude  of  the  place  will,  of  course,  never  sink  below 
the  horizon, — the  radius  of  the  "Circle  of  Perpetual  Apparition ," 


P' 


T' 

FIG.  7.  —  The  Oblique  Sphere  and  Diurnal  Circles. 

as  it  is  called  (the  shaded  cap  around  P  in  the  figure),  being  just 
equal  to  the  height  of  the  pole,  and  becoming  larger  as  the  latitude 
increases.  On  the  other  hand,  stars  within  the  same  distance  of  the 
depressed  pole  will  lie  within  the  "  Circle  of  Perpetual  Occultation," 
and  will  never  rise  above  the  horizon. 

A  star  exactly  on  the  celestial  equator  will  have  its  diurnal  circle 
EQ  WQ'  bisected  by  the  horizon,  and  will  be  above  the  horizon  just 
as  long  as  below  it.  A  star  north  of  the  equator  (if  the  North  Pole 
is  the  elevated  one)  will  have  more  than  half  of  its  diurnal  circle 
above  the  horizon,  and  will  be  visible  more  than  half  the  time  ;  as,  for 
instance,  a  star  at  A :  and  of  course  the  reverse  will  be  true  of  stars 


20  DEFINITIONS   AND    GENERAL   CONSIDERATIONS. 

on  the  other  side  of  the  equator.1  Whenever  the  sun  is  north  of 
the  equator,  the  day  will  therefore  be  longer  than  the  night  for  all 
stations  in  northern  latitude  :  how  much  longer  will  depend  both 
on  the  latitude  of  the  place  and  the  sun's  distance  from  the  celestial 
equator. 


1 A  Celestial  Globe  will  be  of  great  assistance  in  studying  these  diurnal  circles. 
The  north  pole  of  the  globe  must  be  elevated  to  an  angle  equal  to  the  latitude  of 
the  observer,  which  can  be  done  by  means  of  .the  degrees  marked  on  the  brass 
meridian.  It  will  then  at  once  be  easily  seen  what  stars  never  set,  which  ones 
never  rise,  and  during  what  part  of  the  24  hours  any  heavenly  body  at  a  known 
distance  from  the  equator  is  above  or  below  the  horizon. 

NOTE  TO  ART.  6. 

The  ordinary  estimate  of  the  apparent  size  of  the  sun  and  moon  as  "  about  a 
foot  in  diameter"  probably  rests  upon  a  physiological  fact,  —  viz.,  that  in  judg- 
ing moderate  distances,  where  we  are  not  helped  by  intervening  objects,  we 
have  to  depend  upon  the  muscular  sensation  of  strain  in  converging  our  eyes 
towards  the  object  looked  at.  For  distances  not  exceeding  fifty  or  sixty  feet 
this  is  fairly  accurate,  but  for  distances  above  a  hundred  feet  it  entirely  fails. 
When,  therefore,  we  look  at  the  moon  in  mid-heaven,  our  eyes  directly  inform 
us  that  it  is  at  least  a  hundred  feet  away  ;  on  the  other  hand,  from  the  absence 
of  intervening  objects  we  instinctively  estimate  the  distance  as  the  least  possible 
consistent  with  the  non-convergence  of  our  eyes,  and  accordingly  imagine  the 
size  of  the  disc  to  be  about  that  of  a  ball  which  at  a  distance  of  a  hundred  feet 
or  so  would  subtend  the  same  angle  of  half  a  degree  ;  le.,  about  a  foot. 


ASTRONOMICAL  INSTRUMENTS.  21 


CHAPTER  II. 

ASTRONOMICAL  INSTRUMENTS. 

34.  ASTRONOMICAL  observations  are  of  various  kinds :  sometimes 
we  desire  to  ascertain  the  apparent  distance  between  two  bodies  at  a 
given  time  ;  sometimes  the  position  which  a  body  occupies  at  a  given 
time,  or  the  moment  it  arrives  at  a  given  circle  of  the  sky,  usually 
the  meridian.     Sometimes  we  wish  merely  to  examine  its.  surface,  to 
measure  its  light,  or  to  investigate  its  spectrum ;  and  for  all  these 
purposes  special  instruments  have  been  devised. 

We  propose  in  this  chapter  to  describe  very  briefly  a  few  of  the 
most  important. 

35.  Telescopes  in  General. — Telescopes  are  of  two  kinds,  refract- 
ing and  reflecting.     The  former  were  first  invented,  and  are  much 
more  used,  but  the  largest  instruments  ever  made  are  reflectors.     Iii 
both  the  fundamental  principle  is  the  same.     The  large  lens,  or  mir- 
ror, of  the  instrument  forms  at  its  focus  a  real  image  of  the  object 
looked  at,  and  this  image  is  then  examined  and  magnified  by  the  eye- 
piece, which  in  principle  is  only  a  magnifying-glass. 

In  the  form  of  telescope,  however,  introduced  by  Galileo,1  and  still  used 
as  the  "  opera-glass,"  the  rays  from  the  object-glass  are  intercepted  by  a  con- 
cave lens  which  performs  the  office  of  an  eye-piece  before  they  meet  at  the 
focus  to  form  the  "real  image."  But  on  account  of  the  smallness  of  the 
field  of  view,  and  other  objections,  this  form  of  telescope  is  never  used  when 
any  considerable  power  is  needed. 

1  In  strictness,  Galileo  did  not  invent  the  telescope.  Its  first  invention 
seems  to  have  been  in  1608,  by  Lipperhey,  a  spectacle-maker  of  Middleburg, 
in  Holland;  though  the  honor  has  also  been  claimed  for  two  or  three  other 
Dutch  opticians.  Galileo,  in  his  "Nuncius  Sydereus,"  published  in  March, 
1610,  himself  says  that  he  had  heard  of  the  Dutch  instruments  in  1609,  and 
by  so  hearing  was  led  to  construct  his  own,  which,  however,  far  excelled  in 
power  any  that  had  been  made  previously ;  and  he  was  the  first  to  apply 
the  telescope  to  Astronomy  See  Grant's  "History  of  Astronomy,"  pp.  614 
and  seqq. 


22  ASTKONOMICAL  INSTRUMENTS. 

36.  Simple  Refracting  Telescope. — This  consists  essentially  as 
shown  in  the  figure  (Fig.  8),  of  a  tube  containing  two  lenses  :  a  single 
convex  lens,  A,  called  the  object-glass  ;  and  another,  of  smaller  size 
and  short  focus,  .B,  called  the  eye-piece.  Recalling  the  principles  of 
lenses  the  student  will  see  that  if  the  instrument  be  directed  at  a  dis- 
tant object,  the  moon,  for  instance,  all  the  rays,  Oo^c^,  which  fall 
upon  the  object-glass  from  a  point  at  the  top  of  the  moon,  will  be 
collected  at  a  in  the  focal  plane,  at  the  bottom  of  the  image.  Simi- 
larly rays  from  the  bottom  of  the  moon  will  go  to  b  at  the  top  of  the 
image ;  moreover,  since  the  rays  that  pass  through  the  optical  centre 
of  the  lens,  o,  are  undeviated,1  the  angle  a0obQ  will  equal  boa ;  or,  in 
other  words,  if  the  focal  length  of  the  lens  be  five  feet,  for  instance, 
then  the  image  of  the  moon,  seen  from  a  distance  of  five  feet,  will 
appear  just  as  large  as  the  moon  itself  does  in  the  sky,  —  it  will 
subtend  the  same  angle.  If  we  look  at  it  from  a  smaller  distance, 


FIG.  8.  —  Path  of  the  Rays  in  the  Astronomical  Telescope. 

say  from  a  distance  of  one  foot,  the  image  will  look  larger  than  the 
moon ;  and  in  fact,  without  using  an  eye-piece  at  all,  a  person  with 
normal  eyes  can  obtain  considerable  magnifying  power  from  the 
object-glass  of  a  large  telescope.  With  a  lens  of  ten  feet  focal 
length,  such  as  is  ordinarily  used  in  an  8-inch  telescope,  one  can 
easily  see  the  mountains  on  the  moon  and  the  satellites  of  Jupiter, 
by  taking  out  the  eye-piece,  and  putting  the  eye  in  the  line  of  vision 
some  eight  or  ten  inches  back  of  the  eye-piece  hole. 

The  image  is  a  real  one;  i.e.,  the  rays  that  come  from  different 
points  in  the  object  actually  meet  at  corresponding  points  in  the  im- 
age, so  that  if  a  photographic  plate  were  inserted  at  «6,  and  prop- 
erly exposed,  a  picture  would  be  obtained. 

If  we  look  at  the  image  with  the  naked  eye,  we  cannot  come  nearer 

1  In  this  explanation,  we  use  the  approximate  theory  of  lenses  (in  which  their 
thickness  is  neglected),  as  given  in  the  elementary  text-books.  The  more  exact 
theory  of  Gauss  and  later  writers  would  require  some  slight  modifications  in  our 
statements,  but  none  of  any  material  importance.  For  a  thorough  discussion, 
see  Jamin,  "  Traite de Physique"  or  Encyc.  Britannica,  —  Optics. 


ASTRONOMICAL  INSTRUMENTS.  23 

to  the  image  (unless  near-sighted)  than  eight  or  ten  inches,  and  so 
cannot  get  any  great  magnifying  power ;  but  if  we  use  a  magnify- 
ing-glass,  we  can  approach  much  closer. 

37.  Magnifying  Power.  — If  the  eye-piece  B  is  set  at  a  distance 
from  the  image  equal  to  its  principal  focal  distance,  then  an3T  pencil  of 
rays  from  any  point  of  the  image  will,  after  passing  the  lens,  be  con- 
verted into  a  parallel  beam,  and  will  appear  to  the  eye  to  come  from 
a  point  at  an  infinite  distance,  as  if  from  an  object  in  the  sky.     The 
rays  which  came  from  the  top  of  the  moon,  for  instance,  and  are  col- 
lected at  a  in  the  image,  will  reach  the  eye  as  a  beam  parallel  to  the 
line  ac,  which  connects  a  with  the  optical  centre  of  the  eye-piece.     Simi- 
larly with  the  rays  which  meet  at  b.     The  observer,  therefore,  will 
see  the  top  of  the  moon's  disc  in  the  direction  cfc,  and  the  bottom  in 
the  direction  cl.     It  will  appear  to  him  inverted,  and  greatly  magni- 
fied ;  its  apparent  diameter,  as  seen  by  the  naked  eye  and  measured 
by  the  angle  aob  (or  its  equal  60oa0),  having  been  increased  to  acb. 
Since  both  these  angles  are  subtended  by  the  same  line  a&,  and  are 
small  (the  figure,  of  course,  is  much  out  of  proportion),  they  must 
be  inversely  proportional  to  the  distance  ob  and  cb  ;  i.e.,  boa :  bca  = 
cb :  ob ;  or,  putting  this  into  words :  The  ratio  between  the  natural 
apparent  diameter  of  the  object,  and  its  diameter  as  seen  through  the 
telescope,  is  equal  to  the  ratio  between  the  focal  lengths  of  the  eye- 
lens   and   object-glass.     This   ratio   is   called   the  magnifying  power 
of   the   telescope,   and   is   therefore   given   by   the   simple   formula 

F 

M—  — ,  where  .Pis  the  focal  length  of  the  object-glass  and  /  that  of 

eye-piece,1  while  M  is  the  magnifying  power. 

If,  for  example,  the  object-glass  have  a  focal  length  of  thirty  feet, 
and  the  eye-piece  of  one  inch,  the  magnifying  power  will  be  360  ;  the 
power  may  be  changed  at  pleasure  by  substituting  different  eye- 
pieces, of  which  every  large  telescope  has  an  extensive  stock. 

38.  Brightness  of  Image.  —  Since  all  the  rays  from  a  star  which 
fall  upon  the  large  object-glass  are  transmitted  to  the  observer's  eye 
(neglecting  the  losses  by  absorption  and  reflection) ,  he  obviously  re- 


1  A  magnifying  power  of  1  is  no  magnifying  power  at  all.  Object  and  image 
subtend  equal  angles.  A  magnifying  power  denoted  by  a  fraction,  say  \,  would 
be  a  minifying  power,  making  the  object  look  smaller,  as  when  we  look  at  an  ob 
ject  through  the  wrong  end  of  a  spy-glass. 


24  ASTRONOMICAL  INSTRUMENTS. 

ceives  a  quantity  of  light  much  greater  than  he  would  naturally  get  i 
as  many  times  greater  as  the  area  of  the  object-lens  is  greater  than 
that  of  the  pupil  of  the  eye.  If  we  estimate  this  latter  as  having  a 
diameter  of  one-fifth  of  an  inch,  then  a  1-inch  telescope  would  in- 
crease the  light  twenty-five  times,  a  10-inch  instrument  2500  times, 
and  the  great  Lick  telescope,  of  thirty-six  inches'  aperture,  32,400 
times,  the  amount  being  proportional  to  the  square  of  the  diameter 
of  the  lens. 

It  must  not  be  supposed,  however,  that  the  apparent  brightness  of 
an  object  like  the  moon,  or  a  planet  which  shows  a  disc,  is  increased 
in  any  such  ratio,  since  the  eye-piece  spreads  out  the  light  to  cover  a 
vastly  more  extensive  angular  area,  according  to  its  magnifying 
power ;  in  fact,  it  can  be  shown  that  no  optical  arrangement  can 
show  an  extended  surface  brighter  than  it  appears  to  the  naked 
eye.  But  the  total  quantity  of  light  utilized  is  greatly  increased 
by  the  telescope,  and  in  consequence,  multitudes  of  stars,  far  too 
faint  to  be  visible  to  the  unassisted  eye,  are  revealed ;  and,  what  is 
practically  very  important,  the  brighter  stars  are  easily  seen  by  day 
with  the  telescope. 

39.  Distinctness  of  Image.  —  This  depends  upon  the  exactness 
with  which  the  lens  gathers  to  a  single  point  in  the  focal  image  all 
the  rays  which  emanate  from  the  corresponding  point  in  the  object. 
A  single  lens,  with  spherical  surfaces,  cannot  do  this  very  perfectly, 
the  "aberrations"  being  of  two  kinds,  the  spherical  aberration  and 
the  chromatic.     The  former  could  be  corrected,  if  it  were  worth  while, 
by  slightly  modifying  the  form  of  the  lens-surfaces ;   but  the  latter, 
which  is  far  more  troublesome,  cannot  be  cured  in  any  such  way. 
The  violet  rays  are  more   refrangible  than  the  red,  and  come  to  a 
focus  nearer  the  lens ;  so  that  the  image  of  a  star  formed  by  such 
a  lens  can  never  be  a  luminous  point,  but  is  a  round  patch  of  light 
of  different  color  at  centre  and  edge. 

40.  Long  Telescopes.  —  By  making  the  diameter  of  the  lens  very  small 
as  compared  with  its  focal  length,  the  aberration  becomes  less  conspicuous  ; 
and  refractors  were  used,  about  1680,  having  a  length  of  more  than  100  feet 
and  a  diameter  of  five  or  six  inches.     The  object-glass  was  mounted  at  the 
top  of  a  high  pole  and  the  eye-piece  was  on  a  separate  stand  below.     Huy- 
ghens  and  Cassini  both  used  such  "  aerial  telescopes,"  and  one  of  Huyghens' 
object-glasses,  of  six  inches  aperture  and  123  feet  focus,  is  still  preserved 
in  the  Museum  of  the  Royal  Society  in  London. 


ASTRONOMICAL   INSTRUMENTS.  25 

41.  The  Achromatic  Telescope. — The  chromatic  aberration  of  a 
lens,  as  has  been  said,  cannot  be  cured  by  any  modification  of  the  lens 
itself;  but  it  was  discovered  in  England  about  1760  that  it  can  be 
nearly  corrected  by  making  the  object-glass  of  two  (or  more)  lenses, 
of  different  kinds  of  glass,  one  of  the  lenses  being  convex  and  the 
other  concave.     The  convex  lens  is  usually  made  of  crown  glass,  the 
concave  of  flint  glass.     At  the  same  time,  by  properly  choosing  the 
curves,  the  spherical  aberration  can  also  be  destroyed,  so  that  such  a 
compound  object-glass  comes  reasonably  near  to  fulfilling  the  con- 
dition, that  it  should  gather  to  a  mathematical  point  in  the  image  all 
the  rays  that  reach  the  object-glass  from  a  single  point  in  the  object. 

These  object-glasses  admit  of  a  considerable  variety  of  forms.  Formerly 
they  were  generally  made,  as  in  Fig.  9,  No.  3,  having  the  two  lenses  close 
together,  and  the  adjacent  surfaces  of  the  same,  or  nearly  the  same,  curva- 
ture. In  small  object-glasses  the  lenses  are  often  cemented  together  with 
Canada  balsam  or  some  other  transparent  medium.  At  present  some  of  the 
best  makers  separate  the  two  lenses  by  a  considerable  distance,  so  as  to 
admit  a  free  circulation  of  air  between  them ;  in  the  Pulkowa  and  Prince- 
ton object-glasses,  constructed  by 
Clark,  the  lenses  are  seven  inches 
apart,  and  in  the  Lick  telescope  six 
and  a  half  inches;  as  in  No.  1.  In 
a  form  devised  by  Gauss  (No.  2),  Clark 

Which  has  some  advantages,  but  is  Fl0'9'- 
difficult  of  construction,  the  curves 
are  very  deep,  and  both  the  lenses  are  of  watch-glass  form  —  concave  on  one 
side  and  convex  on  the  other.  In  all  these  forms  the  crown  glass  is  outside ; 
Steinheil,  Hastings,  and  others  have  constructed  lenses  with  the  flint-glass 
lens  outside.  Object-glasses  are  sometimes  made  with  three  lenses  instead 
of  two ;  a  slightly  better  correction  of  aberrations  can  be  obtained  in  this 
way,  but  the  gain  is  too  small  to  pay  for  the  extra  expense  and  loss  of  light. 

42.  Secondary  Spectrum.  —  It  is  not,  however,  possible  with  the 
kinds  of  glass  ordinarily  employed  to  secure  a  perfect  correction  of 
the  color.     Our  best  achromatic  lenses  bring  the  yellowish  green 
rays  to  a  focus  nearer  the  lens  than  they  do  the  red  and  violet.     In 
consequence,  the  image  of  a  bright  star  is  surrounded  by  a  purple 
halo,  which  is  not  very  noticeable  in  a  good  telescope  of  small  size, 
but  is  very  conspicuous  and  troublesome  in  a  large  instrument. 

This  imperfection  of  achromatism  makes  it  unsatisfactory  to  use  an  ordi- 
nary lens  (visually  corrected)  for  astronomical  photography.  To  fit  it  to 
make  good  photographs,  it  must  either  be  specially  corrected  for  the  rays 


26  ASTRONOMICAL  INSTRUMENTS. 

that  are  most  effective  in  photography,  the  blue  and  violet  (in  which  case  it 
will  be  almost  worthless  visually),  or  else  a  subsidiary  lens,  known  as  a  "pho- 
tographic corrector,"  may  be  provided,  which  can  be  put  on  in  front  of  the 
object-glass  when  needed.  A  new  form  of  object-glass,  devised  independ- 
ently by  Pickering  in  this  country  and  Stokes  in  England,  avoids  the  necessity 
of  a  third  lens  by  making  the  crown-glass  lens  of  such  a  form  that  when 
put  close  to  the  flint  lens,  with  the  flatter  side  out,  it  makes  a  perfect  object- 
glass  for  visual  purposes ;  but  by  simply  reversing  the  crown  lens,  with  the 
more  convex  side  outward,  and  separating  the  lenses  an  inch  or  two,  it  be- 
comes a  photographic  object-glass. 

42*.  Photo-visual  Objectives.  —  Much  is  hoped  from  the  new  kinds  of 
glass  now  made  at  Jena,  but  there  has  been  great  difficulty  in  producing  discs 
satisfactorily  homogeneous,  of  such  chemical  composition  that  the  surfaces 
will  not  "  rust,"  and  large  enough  for  telescopes  of  any  size.  Since  1894, 
however,  the  English  opticians,  Cooke  &  Sons,  have  been  advertising  "  per- 
fectly achromatic  "  triple  object-glasses,  which  are  asserted  to  be  equally 
perfect  for  visual  and  photographic  use.  They  offer  to  make  lenses  twenty 
inches  in  diameter,  but  up  to  1896  bad  produced  only  a  few  as  large  as  six 
or  eight  inches,  which  have  been  examined  and  very  favorably  reported  on 
by  eminent  astronomers.  Possibly  the  new  century  will  open  a  new  era  in 
telescope-making. 

43.  Diffraction  and  Spurious  Disc.  — Even  if  a  lens  were  perfect 
as  regards  the  correction  of  aberrations,  the  "wave"  nature  of  light 
prevents  the  image  of  a  luminous  point  from  being  also  a  point ;  the 
image  must  necessarily  consist  of  a  central  disc,  brightest  in  the  cen- 
tre and  fading  to  darkness  at  the  edge,  and  this  is  surrounded  by  a 
series  of  bright  rings,  of  which,  however,  only  the  smallest  one  is 
generally  easily  seen.  The  size  of  this  disc-and-ring  system  can  be 
calculated  from  the  known  wave-lengths  of  light  and  the  dimensions 
of  the  lens,  and  the  results  agree  very  precisely  with  observation. 
The  diameter  of  the  "  spurious  disc"  varies  inversely  with  the  aper- 
ture of  the  telescope.  According  to  Dawes,  it  is  about  4". 5  for  a 
1-inch  telescope ;  and  consequently  1"  for  a  4£-inch  instrument,  0".5 
for  a  9-inch,  and  so  on. 

This  circumstance  has  much  to  do  with  the  superiority  of  large  instru- 
ments in  showing  minute  details.  No  increase  of  magnifying  power  on  a 
small  telescope  can  exhibit  things  as  sharply  as  the  same  power  on  the  larger 
one ;  provided,  of  course,  that  the  larger  object-glass  is  equally  perfect  in 
workmanship,  and  the  air  in  good  optical  condition. 

If  the  telescope  is  a  good  one,  and  if  the  air  is  perfectly  steady,  —  which 
unfortunately  is  seldom  the  case3  —  the  apparent  disc  of  a  star  should  be 


ASTRONOMICAL   INSTRUMENTS.  27 

perfectly  round  and  well  defined,  without  wings  or  tails  of  any  kind,  having 
around  it  from  one  to  three  bright  rings,  separated  by  distances  somewhat 
greater  than  the  diameter  of  the  disc.  If,  however,  the  magnifying  power 
is  more  than  about  50  to  the  inch  of  aperture,  the  edge  of  the  disc  will  begin 
to  appear  hazy.  There  is  seldom  any  advantage  in  the  use  of  a  magnifying 
power  exceeding  75  to  the  inch,  and  for  most  purposes  powers  ranging  from 
20  to  40  to  the  inch  are  most  satisfactory. 

44.  Eye-Pieces.  — For  many  purposes,  as  for  instance  the  examina- 
tion of  close  double  stars,  there  is  no  better  eye-piece  than  the  simple 
convex  lens  ;  but  it  performs  well  only  when  the  object  is  exactly  in 
the  centre  of  the  field.     Usually  it  is  best  to  employ  for  the  eye-piece 
a  combination  of  two  or  more  lenses. 

Eye-pieces  belong  to  two  classes,  the  positive  and  the  negative.  The 
former,  which  are  much  more  generally  useful,  act  as  simple  magnify- 
ing-glasses,  and  can  be  used  as  hand  magnifiers  if  desired.  The  focal 
image  formed  by  the  object-glass  lies  outside  of  the  eye-piece. 

In  the  negative  eye-pieces,  on  the  other  hand,  the  rays  from  the 
object-glass  are  intercepted  before  they  come  to  the  focus,  and  the 
image  is  formed  between  the  lenses  of  the  eye-piece.  Such  an  eye- 
piece cannot  be  used  as  a  hand  magnifier. 

45.  The  simplest  and  most  common  forms  of  these  eye-pieces  are  the 
Ramsden    (positive)   and 

Huyghenian    (negative). 
Each  is  composed  of  two 

plano-convex    lenses,    but          ^^  Huyghenian 

the      arrangement      and         (Positive)  (Negative) 

curves  differ,  as  shown 
in  Fig.  10.  The  former 
gives  a  very  flat  field  of 
view,  but  is  not  achro- 
matic; the  latter  is  more  FIG.  10. —Various  Forms  of  Telescope  Eye-piece, 
nearly  achromatic,  and 

possibly  defines  a  little  better  just  at  the  centre  of  the  field;  but  the  fact 
that  it  is  a  negative  eye-piece  greatly  restricts  its  usefulness.  In  the  Rams- 
den  eye-piece  the  focal  lengths  of  the  two  component  lenses,  both  of  which 
have  their  flat  sides  out,  are  about  equal  to  each  other,  and  their  distance  is 
about  one-third  of  the  sum  of  the  focal  lengths.  In  the  Huyghenian  the 
curved  sides  of  the  lenses  are  both  turned  towards  the  object-glass;  the 
focal  distance  of  the  field  lens  should  be  exactly  three  times  that  of  the  lens 
next  the  eye,  and  the  distance  between  the  lenses  one-half  the  sum  of  the 
focal  lengths. 

There  are  numerous  other  forms  of  eye-piece,  each  with  its  own  advan- 
tages and  disadvantages.  The  erecting  eye-piece,  used  in  spy-glasses,  is 


28  ASTRONOMICAL   INSTRUMENTS. 

essentially  a  compound  microscope,  and  gives  erect  vision  by  again  invert- 
ing the  already  inverted  image  formed  by  the  object-glass. 

It  is  obvious  that  in  a  telescope  of  any  size  the  object-glass  is  the  most 
important  and  expensive  part  of  the  instrument.  Its  cost  varies  from  a  few 
hundred  dollars  to  many  thousands,  while  the  eye-pieces  generally  cost  only 
from  $5  to  $20  apiece. 

46.  Reticle.  —  When  a  telescope  is  used  for  pointing,  as  in  most 
astronomical  instruments,  it  must  be  provided  with  a  reticle  of  some 
sort.     This  is  usually  a  metallic  frame  with  spider  lines  stretched 
across  it,  placed,  not  near  the  object-glass  itself  (as  is  often  sup- 
posed), but  at  the  focus  of  the  object-glass,  where  the  image  is 
formed,  as  at  a  b  in  Fig.  8. 

It  is  usually  so  arranged  that  it  can  be  moved  in  or  out  a  little  to  get  it 
exactly  into  the  focal  plane,  and  then,  when  the  eye-piece  (positive)  is  ad- 
justed for  the  observer's  eye  to  give  distinct  vision  of  the  object,  the  "wires," 
as  they  are  called,  will  also  be  equally  distinct.  As  spider-threads  are  very 
fragile,  and  likely  to  get  broken  and  displaced,  it  is  often  better  to  substitute 
filaments  of  quartz,  or  a  thin  plate  of  glass  with  lines  ruled  upon  it  and 
blackened.  The  field  of  view,  or  the  threads  themselves,  must  be  illuminated 
in  order  to  make  them  visible  in  darkness. 

47.  The  Reflecting  Telescope.  —  When  the  chromatic  aberration 
of  lenses  came  to  be  understood  through  the  optical  discovery  of 
the  dispersion  of  light  by  Newton,  the  reflecting  telescope  was  in- 
vented, and  held  its  place  as  the  instrument  for  star-gazing  until 
well  into  the  present  century,  when  large  achromatics  began  to  be 
made.     There  are  several  varieties  of  reflecting  telescope,  all  agree- 
ing in  the  substitution  of  a  large  concave  mirror  in  place  of  the 
object-glass  of  the  refractor,  but  differing  in  the  way  in  which  they 
get  at  the  image  formed  by  this  mirror   at  its  focus  in  order  to 
examine  it  with  the  eye-piece. 

48.  In  the  Herschelian  form,  which  is  the  simplest,  but  only  suited  to 
very  large  instruments,  the  mirror  is  tipped  a  little,  so  as  to  throw  the  image 
to  the  side  of  the  tube,  and  the  observer  stands  with  his  back  to  the  object 
and  looks  down  into  the  tube.     If  the  telescope  is  as  much  as  two  or  three 
feet  in  diameter,  his  head  will  not  intercept  enough  light  to  do  much  harm, 
—  not  nearly  so  much  as  would  be  lost  by  the  second  reflection  necessary  in 
the  other  forms  of  the  instrument.     But  the  inclination  of  the  mirror,  and 
the  heat  from  the  observer's  person,  are  fatal  to  any  very  accurate  definition, 
and  unfit  this  form  of  instrument  for  anything  but  the  observation  of  nebulae 
and  objects  which  mainly  require  light-gathering  power. 


ASTRONOMICAL    INSTRUMENTS.  29 

In  the  Newtonian  telescope,  a  small  plane  reflector  standing  at  an  angle 
of  45°  is  placed  in  the  centre  of  the  tube,  so  as  to  intercept  the  rays  reflected 
by  the  large  mirror  a  little  before  they  come  to  their  focus,  and  throw  them 
to  the  side  of  the  tube,  where  the  eye-piece  is  placed. 

In  the  Gregorian  form  (which  was  the  first  invented),  the  large  mirror  is 
pierced  through  its  centre,  and  the  rays  from  it  are  reflected  through  the 
hole  by  a  small  concave  mirror,  placed  a  little  outside  of  the  principal  focus 
at  the  mouth  of  the  tube.  With  this  instrument  one  looks  directly  at  the 
stars  as  with  a  refractor,  and  the  image  is  erect. 

The  Cassegrainian  form  is  very  similar,  except  that  the  small  concave 
mirror  of  the  Gregorian  is  replaced  by  a  convex  mirror,  placed  a  little  inside 
the  focus  of  the  large  mirror,  which  makes  the  instrument  a  little  shorter, 
and  gives  a  flatter  field  of 


Formerly      the      great     j «  _g 

mirror  was  always  made       |  --^~-rj-j---.      5 

of  a  composition  of  cop- 
per and  tin  (two  parts  of 
copper    to    one    of     tin) 
known      as      "  speculum 
metal."      At    present    it 
is  usually  made  of  glass 
silvered  on  the  front  sur- 
face, by  a  chemical  pro- 
cess  which    deposits 
metal  in  a  thin,  brilliant 
film.       These      silver-on- 
glass      reflectors,      when         FlG  n  _  Different  Forms  of  Reflecting  Telescope. 
new,    reflect    much    more     i.  Tne  Herschelian  ;  2.  The  Newtonian  ;  3.  The  Gregorian, 
light  than  the  old  specula, 

but  the  film  does  not  retain  its  polish  so  long.  It  is,  however,  a  comparatively 
simple  matter  to  renew  the  film  when  necessary. 

The  largest  telescopes  ever  made  have  been  reflectors.  At  the  head  of  the 
list  stands  the  enormous  instrument  of  Lord  Rosse,  constructed  in  1842,  with 
a  mirror  six  feet  in  diameter  and  sixty  feet  focal  length.  Next  in  order 
comes  the  five-foot  silver-on-glass  reflector  of  Mr.  Common1  (1889),  and 
another  of  the  same  size,  figured  by  Mr.  Ritchey,  and  recently  mounted 
at  the  Mount  Wilson  Solar  Observatory,  near  Pasadena,  California.  Then 
there  are  several  instruments  of  four  feet  aperture,  first  among  which  is 
the  great  telescope  of  the  elder  Herschel,  built  in  1789. 

49.  Relative  Advantages  of  Refractors  and  Reflectors There  has 

been  a  good  deal  of  discussion  on  this  point,  and  each  construction  has  its 
partisans. 

In  favor  of  the  reflectors  we  may  mention,  — 

First.    Ease  of  construction  and  consequent  cheapness.    The  concave  mirror 

1  Acquired  and  mounted  by  Harvard  College  Observatory  in  1905. 


30  ASTRONOMICAL    INSTRUMENTS. 

has  but  one  surface  to  figure  and  polish,  while  an  object-glass  has  four. 
Moreover,  as  the  light  goes  through  an  object-glass,  it  is  evident  that  the 
glass  employed  must  be  perfectly  clear  and  of  uniform  density  through  and 
through  ;  while  in  the  case  of  the  mirror,  the  light  does  not  penetrate  the 
material  at  all.  This  makes  it  vastly  easier  to  get  the  material  for  a  large 
mirror  than  for  a  large  lens. 

Second  (and  immediately  connected  with  the  preceding).  The  possibility 
of  making  reflectors  much  larger  than  refractors.  Lord  Kosse's  great  reflector 
is  six  feet  in  diameter,  while  the  Yerkes  telescope,  the  largest  refractor  in 
use,  is  only  forty  inches.  (Its  focal  length,  however,  is  sixty-five  feet.) 

Third.  Perfect  achromatism.  This  is  unquestionably  a  very  great  ad- 
vantage, especially  in  photographic  and  spectroscopic  work. 

But,  on  the  whole,  the  advantages  are  generally  considered  to  lie  with 
the  refractors. 

In  their  favor  we  mention  :  — 

First.     Great  superiority  in  light.     No  mirror  (unless,  perhaps,  a  freshly 

polished  silver-on-glass  film)  reflects  much 
more  than  three-quarters  of  the  incident 
light ;  while  a  good  (single)  lens  trans- 
mits over  90  per  cent.  In  a  good  refrac- 
tor about  80  per  cent  of  the  light  reaches 
the  eye,  after  passing  through  the  four 
lenses  of  the  object-glass  and  eye-piece. 
In  a  Newtonian  reflector,  in  average 
condition,  the  percentage  seldom  exceeds 
50  per  cent,  and  more  frequently  is  lower 
than  higher. 

Second.   Better  definition.  —  Any  slight 

error  at  a  point  in  the  surface  of  a  glass 
FIG.  12. —  Effect  of  Surface  Errors  in  a     .  •    .,•£.  j  T.     £     ,,  ^ 

Mirror  and  in  a  Lens.  lens>  whether  caused  by  faulty  workman- 

ship or  by  distortion,  affects  the  direction 

of  the  ray  passing  through  it  only  one-third  as  much  as  the  same  error  on 
the  surface  of  a  mirror  would  do. 

If,  for  instance,  in  Fig.  12,  an  element  of  the  surface  at  P  is  turned  out 
of  its  proper  direction,  aa',  by  a  small  angle,  so  as  to  take  the  direction  bb', 
then  the  reflected  ray  will  be  sent  to  /,  and  its  deviation  will  be  twice  the 
angle  dPb.  But  since  the  index  of  refraction  of  glass  is  about  1.5  the 
change  in  the  direction  of  the  refracted  ray  from  R  to  r  will  only  be  about 
two-thirds  of  aPb. 

Moreover,  so  far  as  distortions  are  concerned,  when  a  lens  bends  a  little 
by  its  own  weight,  both  sides  are  affected  in  a  nearly  compensatory  manner, 
while  in  a  mirror  there  is  no  such  compensation.  As  a  consequence,  mirrors 
very  seldom  indeed  give  any  such  definition  as  lenses  do.  The  least  fault 
of  workmanship,  the  least  distortion  by  their  own  weight,  the  slightest  dif- 
ference of  temperature,  between  front  and  back,  will  absolutely  ruin  the 
image,  while  a  lens  would  be  but  slightly  affected  in  its  performance  by  the 
same  circumstances. 


ASTRONOMICAL  INSTRUMENTS.  31 

Third.  Permanence.  The  lens,  once  made,  and  fairly  taken  care  of, 
suffers  no  deterioration  from  age ;  but  the  metallic  speculum  or  the  silver 
film  soon  tarnishes,  and  must  be  repolished  every  few  years.  This  alone 
is  decisive  in  most  cases,  and  relegates  the  reflector  mainly  to  the  use  of 
those  who  are  themselves  able  to  construct  their  own  instruments. 

To  these  considerations  we  may  add  that  a  refractor,  though  more  expen- 
sive than  a  reflector  of  similar  power,  is  not  only  more  permanent,  and  less 
likely  to  have  its  performance  affected  by  accidental  circumstances,  but  is 
lighter  and  more  convenient  to  use. 

s  50.  Time-Keepers  and  Time-Recorders.  —  The  Clock,  Chronometer, 
and  Chronograph.  —  Modern  practical  astronomy  owes  its  develop- 
ment as  much  to  the  clock  and  chronometer  as  to  the  telescope.  The 
ancients  possessed  no  accurate  instruments  for  the  measurement  of 
time,  and  until  within  200  years,  the  only  reasonably  precise  method 
of  fixing  the  time  of  an  important  observation,  as,  for  instance,  of 
an  eclipse,  was  by  noting  the  altitude  of  the  sun,  or  of  some  known 
star  at  or  very  near  the  moment. 

It  is  true  that  the  Arabian  astronomer  Ibn  Jounis  had  made  some 
use  of  the  pendulum  about  the  year  1000  A.D.,  more  than  500  years 
before  Galileo  introduced  it  to  Europeans.  But  it  was  not  until 
nearly  a  century  after  Galileo's  discovery  that  Huyghens  applied  it 
to  the  construction  of  clocks  (in  1657). 

So  far  as  the  principles  of  construction  are  concerned,  there  is  no 
difference  between  an  astronomical  clock  and  any  other.  As  a  matter 
of  convenience,  however,  the  astronomical  clock  is  almost  invariably 
made  to  beat  seconds  (rarely  half-seconds),  and  has  a  conspicuous 
second-hand,  while  the  hour-hand  makes  but  one  revolution  a  day, 
instead  of  two,  as  usual,  and  the  face  is  marked  for  twenty-four  hours 
instead  of  twelve.  Of  course  it  is  constructed  with  extreme  care  in 
all  respects. 

The  Escapement,  or  "Scapement,"  is  often  of  the  form  known  as  the  "  Graham 
Dead-beat";  but  it  is  also  frequently  one  of  the  numerous  " gravity " escape- 
ments which  have  been  invented  by  ingenious  mechanicians.  The  office  of 
the  escapement  is  to  be  "  unlocked  "  by  the  pendulum  at  each  vibration,  so 
as  to  permit  the  wheel-work  to  advance  one  step,  marking  a  second  (or  some- 
times two  seconds),  upon  the  clock-face ;  while,  at  the  same  time,  the  escape- 
ment gives  the  pendulum  a  slight  impulse,  just  equal  to  the  resistance  it  has 
suffered  in  performing  the  unlocking.  The  work  done  by  the  pendulum  in 
"unlocking"  the  train,  and  the  corresponding  impulse,  ought  to  be  perfectly 
constant,  in  spite  of  all  changes  in  the  condition  of  the  train  of  wheels ;  and 
it  is  desirable,  though  not  essential,  that  this  work  should  be  as  small  as 
possible. 


32 


ASTRONOMICAL  INSTRUMENTS. 


51.  The  pendulum  itself  is  usually  suspended  by  a  flat  spring,  and 
great  pains  should  be  taken  to  have  the  support  extremely  firm :  this 
;s  often  neglected,  and  the  clock  then  cannot  perform  well. 

Compensation  for  Temperature.  —  In  order  to  keep  perfect  time, 
the  pendulum  must  be  a  "  compensation  pendulum"  ;  i.e.,  constructed 
in  such  a  way  that  changes  of  temperature  will 
not  change  its  length. 

An  uncompeusated  pendulum,  with  steel  rod, 
changes  its  daily  rate  about  one-third  of  a  second 
for  each  degree  of  temperature  (centigrade). 
A  wooden  pendulum  rod  is  much  less  affected 
by  temperature,  but  is  very  apt  to  be  disturbed 
by  changes  of  moisture. 

Graham's  mercurial  pendulum  (Fig.  13)  is  the 
one  most  commonly  used.  It  consists  simply  of  a 
jar  (usually  steel),  three  or  four  inches  in  diameter, 
and  about  eight  inches  high,  containing  forty  or  fifty 
pounds  of  mercury,  and  suspended  at  the  end  of  a 
steel  rod.  When  the  temperature  rises,  the  rod 
lengthens  (which  would  make  the  clock  go  slower)  ; 
but,  at  the  same  time,  the  mercury  expands,  from 
the  bottom  upwards,  just  enough  to  compensate. 
This  pendulum  will  perform  well  only  when  not 
exposed  to  rapid  changes  of  temperature.  Under 
rapid  changes  the  compensation  lags.  If,  for  in- 
stance, it  grows  warm  quickly,  the  rod  will  expand 
before  the  mercury  does ;  so  that,  while  the  mercury  is 
growing  warmer,  the  clock  will  run  slow,  though  after 
it  has  become  warm  the  rate  may  be  all  right. 

A  compensation  pendulum,  constructed  on  the 
principle  of  the  old  gridiron  pendulum  of  Harrison, 
but  of  zinc  and  steel  instead  of  brass  and  steel,  is 
now  much  used.  The  compensation  is  not  so  easily 

adjusted  as  in  the  mercurial  pendulum,  but  when  properly  made  the  mechan- 
ism acts  well,  and  bears  rapid  alterations  of  temperature  much  better  than 
the  mercurial  pendulum.  The  heavy  pendulum-bob,  a  lead  cylinder,  is  hung 
at  the  end  of  a  steel  rod,  which  is  suspended  from  the  top  of  a  zinc  tube, 
and  hangs  through  the  centre  of  it.  This  tube  is  itself  supported  at  the  bottom 
by  three  or  four  steel  rods  which  hang  from  a  piece  attached  to  the  pendu- 
lum spring.  The  standard  clock  at  Greenwich  has  a  pendulum  of  this  kind. 


FIG.  13. 
Compensation    Pendulums. 

1.  Graham's  Pendulum. 

2.  Zinc-Steel  Pendulum. 


52.     Effect  of  Atmospheric  Pressure. — In   consequence  of    the 
buoyancy  of  the  air,  and  its  resistance  to  motion,  a  pendulum  swings 


ASTRONOMICAL  INSTRUMENTS.  33 

a  little  more  slowly  than  it  would  in  vacua ,  and  every  change  in  the 
density  of  the  air  affects  its  rate  more  or  less.  With  mercurial 
pendulums,  of  ordinary  construction,  the  "  barometric  coefficient," 
as  it  is  called,  is  about  one-third  of  a  second  for  an  inch  of  the 
barometer;  i.e.,  an  increase  of  atmospheric  density  which  would 
raise  the  barometer  one  inch  would  make  the  clock  lose  about  one- 
third  of  a  second  daily.  It  varies  considerably,  however,  with  differ- 
ent pendulums. 

It  is  not  very  usual  to  take  any  notice  of  this  slight  disturbance ;  but 
when  the  extremest  accuracy  of  time-keeping  is  aimed  at,  the  clock  is  either 
sealed  in  an  air-tight  case  from  which  the  air  is  partially  exhausted  (as  at 
Berlin),  or  else  some  special  mechanism,  controlled  by  a  barometer,  is  de- 
vised to  compensate  for  the  barometric  changes,  as  at  Greenwich.  In  the 
Greenwich  clock  a  magnet  is  raised  or  lowered  by  the  rise  or  fall  of  the 
mercury  in  a  barometer  attached  to  the  clock-case.  When  the  magnet  rises, 
it  approaches  a  bit  of  iron  two  or  three  inches  above  it,  fixed  to  the  bottom 
of  the  pendulum,  and  the  increase  of  attraction  accelerates  the  rate  just 
enough  to  balance  the  retardation  due  to  the  air's  increased  density  and 
viscosity.  There  are  several  other  contrivances  for  the  same  purpose. 

53.  Error  and  Rate.  —  The  "  error"  or  "  correction"  of  a  clock 
is  the  amount  that  must  be  added  to  the  indication  of  the  clock-face 
at  any  moment  in  order  to  give  the  true  time;  it  is,  therefore,  plus 
(  +  )  when  the  clock  is  slow,  and  minus  ( — )  when  it  is  fast.  The 
rate  of  a  clock  is  the  amount  of  its  daily  gain  or  loss;  plus  (-{-)  when 
the  clock  is  losing.  Sometimes  the  hourly  rate  is  used,  but  "  hourly  " 
is  then  always  specified. 

A  perfect  clock  is  one  that  has  a  constant  rate,  whether  that  rate 
be  large  or  small.  It  is  desirable,  for  convenience'  sake,  that  both 
error  and  rate  should  be  small ;  but  this  is  a  mere  matter  of  adjust- 
ment by  the  user  gf  the  clock,  who  adjusts  the  error  by  setting  the 
hands,  and  the  rate  by  raising  or  lowering  the  pendulum-bob. 

The  final  adjustment  of  rate  is  often  obtained  by  first  setting  the  pendu- 
lum-bob so  that  the  clock  will  run  slow  a  second  or  two  daily,  and  then 
putting  on  the  top  of  the  bob  little  weights  of  a  gramme  or  two,  which  will 
accelerate  the  motion.  They  can  be  dropped  into  place  or  knocked  off  with- 
out stopping  the  clock  or  perceptibly  disturbing  it. 

The  very  best  clocks  will  run  three  or  four  years  without  being  stopped 
for  cleaning,  and  will  retain  their  rate  without  a  change  of  more  than  one- 
fifth  of  a  second,  one  way  or  the  other,  during  the  whole  time.  But  this  is 


34 


ASTRONOMICAL    INSTRUMENTS. 


exceptional  performance.  In  a  run  as  long  as  that,  most  clocks  would  be 
liable  to  change  their  rate  as  much  as  half  a  second  or  more,  and  to  do  it 
somewhat  irregularly. 

54.  The  Chronometer.  —  The  pendulum-clock  not  being  portable, 
it  is  necessary  to  provide  time-keepers  that  are.  The  chronometer 
is  merely  a  carefully  made  watch,  with  a  balance  wheel  compensated 
to  run,  as  nearly  as  possible,  at  the  same  rate  in  different  tempera- 
tures, and  with  a  peculiar  escapement,  which,  though  unsuited  to 
watches  exposed  to  ordinary  rough  usage,  gives  better  results  than 
any  other  when  treated  carefully. 


Fi<*.  14.  —  A  Chronograph  by  Warner  and  Swasey.  ^ 

The  box-chronometer  used  on  ship-board  is  usually  about  twice  the  diameter 
of  a  common  pocket  watch,  and  is  mounted  on  gimbals,  so  as  to  keep  hori- 
zontal at  all  times,  notwithstanding  the  motion  of  the  vessel.  It  usually 
beats  half-seconds.  It  is  not  possible  to  secure  in  the  chronometer-balance 
as  perfect  a  temperature  correction  as  in  the  pendulum.  For  this  and  other 


ASTRONOMICAL    INSTRUMENTS.  35 

reasons  the  best  chronometers  cannot  quite  compete  with  the  best  clocks  in 
precision  of  time-keeping ;  but  they  are  sufficiently  accurate  for  most  pur- 
poses, and  of  course  are  vastly  more  convenient  for  field  operations.  They 
are  simply  indispensable  at  sea.  Never  turn  the  hands  of  a  chronometer 
backward. 

55.  Before  the  invention  of  the  telegraph  it  was  customary  to 
note  time  merely  "  by  eye  and  ear.'7      The  observer,  keeping  his 
time-piece  near  him,  listened  to  the  clock-beats,  and  estimated  as 
closely  as  he  could,  in  seconds  and  tenths  of  seconds,  the  moment 
when  the  phenomenon  he  was  watching  occurred  —  the  moment,  for 
instance,  when  a  star  passed  across  a  wire  in  the  reticle  of  his  tele- 
scope.    At  present  the  record  is  usually  made  by  simply  pressing 
a  "  key  "  in  the  hand  of  the  observer,  and  this,  by  a  telegraphic 
connection,  makes  a  mark  upon  a  strip  or  sheet  of  paper,  which  is 
moved  at  a  uniform  rate  by  clock-work,  and  graduated  by  seconds- 
signals  from  the  clock  or  chronometer. 

56.  The  Chronograph.  —  This  is  the  instrument  which  carries  the 
marking-pen  and  moves  the  paper  on  which  the  time-record  is  made. 

X  3h  25m  W.O 

50s  H > 

A  /\      A  A f\ 


FIG.  15.  —  Part  of  a  Chronograph  Record. 

The  paper  is  wrapped  upon  a  cylinder,  six  or  seven  inches  in  diameter, 
and  fifteen  or  sixteen  inches  long.  This  cylinder  is  made  to  revolve 
once  a  minute,  by  clock-work,  while  the  pen  rests  lightly  upon  the 
paper  and  is  slowly  drawn  along  by  a  screw-motion,  so  that  it  marks 
a  continuous  spiral.  The  pen  is  carried  on  the  armature  of  an  electro- 
magnet, which  every  other  second  (or  sometimes  every  second)  re- 
ceives a  momentary  current  from  the  clock,  causing  it  to  make  a 
mark  like  those  which  break  the  lines  in  the  figure  annexed. 

The  beginning  of  a  new  minute  (the  60th  sec.)  is  indicated  either 
by  a  double  mark  as  shown,  or  by  the  omission  of  a  mark.  When 
the  observer  touches  his  key  he  also  sends  a  current  through  the 
magnet,  and  thus  interpolates  a  mark  of  his  own  on  the  record,  as 
at  Xin  the  figure  :  the  beginning  of  the  mark  is  the  instant  noted  — 
in  this  case  54.9s.  Of  course  the  minutes  when  the  chronograph  was 
started  and  stopped  are  noted  by  the  observer  on  the  sheet,  and  so 
enable  him  to  identify  the  minutes  and  seconds  all  through  the  record. 


36  ASTRONOMICAL  INSTRUMENTS. 

Many  European  observatories  use  chronographs  in  which  the  record  is 
made  upon  a  long  fillet  of  paper,  instead  of  a  sheet  on  a  cylinder.  The 
instrument  is  lighter  and  cheaper  than  the  American  form,  but  much  less 
convenient. 

The  regulator  of  the  clock-work  must  be  a  "  continuous  "  regulator,  work- 
ing continuously,  and  not  by  beats  like  a  clock-escapement.  There  are 
various  forms,  most  of  which  are  centrifugal  governors,  acting  either  by 
friction  (like  the  one  in  the  figure)  or  by  the  resistance  of  the  air ;  or  else 
"spring-governors,"  in  which  the  motion  of  a  train,  with  a  pretty  heavy 
fly-wheel,  is  slightly  checked  at  regular  intervals  by  a  pendulum. 

57.  Clock-Breaks.  —  The  arrangements  by  which  the  clock  is  made  to 
send  regular  electric  signals  are  also  various.     One  of  the  earliest  and  simplest 
is  a  fine  platinum  wire  attached  to  the  pendulum,  which  swings  through  a 
drop  of  mercury  at  each  vibration.     All  of  the  arrangements,  however,  in 
which  the  pendulum  itself  has  to  make  the  electric  contact  are  objectionable, 
and  for  clocks  using  the  Graham  dead-beat  scapement  no  absolutely  satis- 
factory means  of  giving  the  signals  has  yet  been  devised.     Clocks  with  the 
gravity  escapements  have  a  decided  advantage  in  this  respect.     Their  wheel- 
work  has  no  direct  action  in  driving  the  pendulum,  and  so  may  be  made  to 
do  any  reasonable  amount  of  outside  work  in  the  way  of  "  key-manipula- 
tion "  without  affecting  the  clock-rate  in  the  least.     Usually  a  wheel  on  the 
axis  of  the  scape-wheel  is  made  to  give  the  electric  signals  by  touching  a 
light  spring  with  one  of  its  teeth  every  other  second. 

Chronometers  are  now  also  fitted  up  in  the  same  way,  to  be  used  with  the 
chronograph. 

The  signals  sent  are  sometimes  "  breaks "  in  a  continuous  current,  and 
sometimes  "  makes  "  in  an  open  circuit.  Usage  varies  in  this  respect,  and 
each  method  has  its  advantages.  The  break-circuit  system  is  a  little  simpler 
in  its  connections,  and  possibly  the  signals  are  a  little  more  sharp,  but  it 
involves  a  much  greater  consumption  of  battery  material,  as  the  current  is 
always  circulating,  except  during  the  momentary  breaks. 

58.  Meridian  Observations. — A  large  proportion  of  all  astronomi- 
cal  observations    are   made  at   the    time  when   the    heavenly  body 
observed  is  crossing  the  meridian,  or  very  near  it.     At  that  moment 
the  effects  of  refraction  and  parallax  (to  be  discussed  hereafter)  are 
a  minimum,  and  as  they  act  only  in  a  vertical  plane,  they  do  not  have 
any  influence  on  the  time  at  which  the  body  crosses  the  meridian. 

59.  The  Transit  Instrument  is  the  instrument  used,  in  connection 
with  a  clock  or  chronometer,  and  often  with  a  chronograph  also,  to 
observe  the  time  of  a  star's  "  transit"  across  the  meridian. 

If  the  error  of  the  (sidereal)  clock  is  known  at  the  moment,  this 
observation  will  determine  the  right  ascension  of  the  body,  which,  it 


ASTRONOMICAL   INSTRUMENTS. 


37 


will  be  remembered,  is  simply  the  sidereal  time  at  which  it  crosses  the 
meridian;  i.e.,  the  number  of  hours,  minutes,  and  seconds  by  which  it 
follows  the  vernal  equinox. 

Vice  versa,  if  the  right  ascension  is  known,  the  error  or  correction 
of  the  clock  will  be  determined. 

The  instrument  (Fig.  16)  consists  essentially  of  a  telescope  mounted 
upon  a  stiff  axis  perpendicular  to  the  telescope  tube.  This  axis  is 
placed  horizontal,  east  and  west,  and  turns  on  pivots  at  its  extremi- 
ties, in  Y-bearings  upon  the  top  of  two  fixed  piers  or  pillars.  A 


FIG.  16.  —  The  Transit  Instrument  (Schematic). 

small  graduated  circle  is  attached,  to  facilitate  "setting"  the  telescope 
at  any  designated  altitude  or  declination. 

The  telescope  carries  at  the  eye-end,  in  the  focal  plane  of  the 
object-glass,  a  reticle  of  some  odd  number  of  vertical  wires, — five 
or  more,  —  one  of  which  is  always  in  the  centre,  and  the  others 
are  usually  placed  at  equal  distances  on  each  side  of  it.  One  or 
two  wires  also  cross  the  field  horizontally. 

If  the  pivots  are  true,  and  the  instrument  accurately  adjusted,  it 
is  evident  that  the  central  vertical  wire  ivill  always  follow  the  meridian 
as  the  instrument  is  turned;  and  the  instant  when  a  star  crosses  this 
wire  will  be  the  true  moment  of  the  star's  meridian  transit.  The 


88 


ASTRONOMICAL   INSTRUMENTS. 


3   2    1 


object  in  having  a  number  of  wires  is,  of  course,  simply  to  gain 
accuracy  by 'taking  the  mean  of  a  number  of  observations  instead  of 
depending  upon  a  single  one. 

In  order  to  "level"  the  axis  properly,  a  delicate  spirit-level  is  an 
essential  adjunct ;  it  is  usual,  also,  (and  important)  to  provide  a  con- 
venient "reversing  apparatus,"  by  which  the  instrument  can  be 

turned  half  round,  making  the  eastern  and 
western  pivots  change  places. 

The  instrument  must  be  thoroughly  stiff 
and  rigid,  without  loose  joints  or  shaky 
screws ;  and  the  two  pivots  must  be  accu- 
rately round,  precisely  in  line  with  each  other, 
free  from  taper,  and  precisely  of  the  same 
size;  all  of  which  conditions  may  be  summed 
up  by  saying  that  they  must  be  portions  of 
one  and  the  same  geometrical  cylinder. 

FIG.  17.  — Reticle  of  the  Transit 
Instrument. 

The  prpper  construction  and  grinding  of 

these  pivots,  which  are  usually  of  hard  bell  metal  (sometimes  of  steel), 
taxes  the  art  of  the  most  skilful  mechanician.  The  level,  also,  is  a  delicate 
instrument,  and  difficult  to  construct. 

Provision  is  made,  of  course,  for  illuminating  the  field  of  view  at  night 
so  as  to  make  the  reticle  wires  visible.  Usually  one  (or  both)  of  the  pivots 
is  pierced,  and  a  lamp  throws  light  through  the  opening  upon  a  small  mirror 
in  the  centre  of  the  tube,  which  reflects  it  down  upon  the  reticle. 

The  Y's  are  used  instead  of  round  bearings,  in  order  to  prevent  any 
rolling  or  shake  of  the  pivots  as  the  instrument  turns. 

Fig.  18  shows  a  modern  transit  instrument  (portable)  as  actually  con- 
structed by  Fauth  &  Co. 

Another  form  of  the  instrument  is  much  used,  which  is  often 
designated  as  the  "  Broken  Transit."  A  reflector  in  the  central  cube 
throws  the  rays  coming  from  the  object-glass,  out  at  right  angles 
through  one  end  of  the  axis,  where  the  eye-piece  is  placed ;  so  that 
the  observer  does  not  have  to  change  his  position  at  all  for  differ- 
ent stars,  but  simply  looks  straight  forward  horizontally.  It  is  very 
convenient  and  rapid  in  actual  work,  but  the  observations  require  a 
considerable  correction  for  flexure  of  the  axis. 

60.  Adjustments.  —  (1)  Focus  and  verticality  of  wires.  •  (2)  Collimation. 
(3)  Level.  (4)  Azimuth. 

First.  The  first  thing  to  do  after  the  instrument  is  set  on  its  supports  and 
the  axis  roughly  levelled,  is  to  adjust  the  reticle.  The  eye-piece  is  drawn  out 


ASTRONOMICAL   INSTRUMENTS. 


39 


FIG.  18.  —  A  3-inch  Transit,  with  reversing  apparatus.    Fauth  &  Co. 

or  pushed  in  until  the  wires  appear  perfectly  sharp,  and  then  the  instrument 
is  directed  to  a  star  or  to  some  distant  object  (not  less  than  a  mile  away), 
and  without  disturbing  the  eye-piece,  the  sliding-tube,  which  carries  the 
reticle,  is  drawn  out  or  pushed  in  until  the  object  is  also  distinct  at  the 


40  ASTRONOMICAL   INSTRUMENTS. 

same  time  with  the  wires.  If  this  adjustment  is  correctly  made,  motion  of 
the  eye  in  front  of  the  eye-piece  will  not  produce  any  apparent  displacement 
of  the  object  in  the  field,  with  reference  to  the  wires.  To  test  the  vertically 
of  the  wires,  the  telescope  is  moved  up  and  down  a  little,  while  looking  at  the 
object ;  if  the  axis  is  level  and  the  wires  vertical,  the  wire  will  not  move  off 
from  the  object  sideways.  There  are  screws  provided  to  turn  the  reticle  a 
little,  so  as  to  effect  this  adjustment. 

When  the  wires  have  been  thus  adjusted  for  focus  and  vertically,  the 
reticle-slide  should  be  tightly  clamped  and  never  disturbed  again.  The  eye- 
piece can  be  moved  in  and  out  at  pleasure,  to  secure  distinct  vision  for  differ- 
ent eyes,  but  it  is  essential  that  the  distance  between  the  object-glass  and  the 
reticle  remain  constant. 

Second.  Collimation.  The  line  joining  the  optical  centre  of  the  object- 
glass  with  the  middle  wire  of  the  reticle  is  called  the  "  line  of  collimation" 
and  this  line  must  be  made  exactly  perpendicular  to  the  axis  of  rotation 
by  moving  the  reticle  slightly  to  one  side  or  the  other  by  means  of  the 
adjusting  screws  provided  for  the  purpose.  The  simplest  way  of  effect- 
ing the  adjustment  is  to  point  the  instrument  on  some  well-defined  dis- 
tant object,  like  a  nail-head  or  a  joint  in  brickwork,  and  then  carefully 
to  "  reverse  "  the  instrument  without  disturbing  the  stand.  If  the  middle 
wire,  after  reversal,  points  just  as  it  did  before,  the  "  collimation  "  is  correct ; 
if  not,  the  middle  wire  must  be  moved  half  way  towards  the  object  by  the 
screws. 

Collimator.  —  It  is  not  always  easy  to  find  a  distant  object  on  which  to 
make  this  adjustment,  and  a  "collimator"  may  be  substituted  with  advantage. 
This  is  simply  a  telescope  mounted  horizontally  on  a  pier  in  front  of  the 
transit  instrument,  so  that  when  the  transit  telescope  is  horizontal,  it  can 
look  straight  into  the  collimator,  which  ought  to  be  of  about  the  same  size 
as  the  transit  itself. 

In  the  focus  of  the  collimator  object-glass  are  placed  two  wires  forming 
an  X,  and  thus  placed  they  can  be  seen  by  a  telescope  looking  into  the  colli- 
mator just  as  distinctly  as  if  they  were  at  an  infinite  distance  and  really  celes- 
tial objects.  The  instrument  furnishes  us  a  mark  optically  celestial,  but 
mechanically  within  reach  of  our  finger-ends  for  illumination,  adjustment, 
etc.  If  the  pier  on  which  it  is  mounted  is  firm,  the  collimator  cross  is  in  all 
respects  as  good  as  a  star,  and  much  more  convenient. 

Third.  Level.  The  adjustment  for  level  is  made  by  setting  a  striding 
level  on  the  pivots  of  the  axis,  reading  the  level,  then  reversing  the  level 
(not  the  transit)  and  reading  it  again.  If  the  pivots  are  round  and  of  the 
same  size,  the  difference  between  the  level-readings  direct  and  reversed  will 
indicate  the  amount  by  which  one  pivot  is  higher  than  the  other.  One  of 
the  Y's  is  made  so  that  it  can  be  raised  and  lowered  slightly  by  means  of  a 
screw,  and  this  gives  the  means  of  making  the  axis  horizontal.  If  the 
pivots  are  not  of  the  same  size  (and  they  never  are  absolutely),  the  astronomer 
must  determine  and  allow  for  the  difference. 


ASTRONOMICAL   INSTRUMENTS.  41 

Fourth.  Azimuth.  In.  order  that  the  instrument  may  indicate  the  meridian 
truly,  its  axis  must  lie  exactly  east  and  west ;  i.e.,  its  azimuth  must  be  90°. 
This  adjustment  must  be  made  by  means  of  observations  upon  the  stars,  and 
is  an  excellent  example  of  the  method  of  successive  approximations,  which 
is  so  characteristic  of  astronomical  investigation,  (a)  After  adjusting  care- 
fully the  focus  and  collimation  of  the  instrument,  we  set  it  north  and  south 
by  guess,  and  level  it  as  precisely  as  possible.  By  looking  at  the  pole  star, 
and  remembering  how  the  pole  itself  lies  with  reference  to  it,  one  can  easily 
set  the  instrument  pretty  nearly;  i.e.,  within  half  a  degree  or  so.  The  middle 
wire  will  now  describe  in  the  sky  a  vertical  circle,  which  crosses  the  meridian 
at  the  zenith,  and  lies  very  near  the  meridian  for  a  considerable  distance 
each  side  of  the  zenith. 

(6)  We  must  next  get  an  "  approximate  "  time ;  i.e.,  set  our  clock  or 
chronometer  nearly  right.  To  do  this,  we  select  from  the  list  of  several 
hundred  stars  in  the  Nautical  Almanac  (which  is  to  be  regarded  in  about 
the  same  light  with  the  clock  and  the  spirit  level,  as  an  indispensable  accessory 
to  the  transit)  a  star  which  is  about  to  cross  the  meridian  near  the  zenith. 
The  difference  between  the  right  ascension  of  the  star  as  given  in  the 
Almanac,  and  the  time  shown  by  the  clock-face,  will  be  very  nearly  the 
error  of  the  clock  at  the  time  of  the  observation  :  not  exactly,  unless  the  dec- 
lination of  the  star  is  such  that  it  passes  exactly  through  the  zenith,  but 
very  nearly,  since  the  star  crosses  the  meridian  near  the  zenith.  We  now 
have  the  time  within  a  second  or  two. 

(c)  Next  turn  down  the  telescope  upon  some  Almanac  star,  which  is 
soon  to  cross  the  meridian  within  10°  of  the  pole.  It  will  appear  to  move 
very  slowly.  A  little  before  the  time  it  should  reach  the  meridian,  move  the 
whole  frame  of  the  instrument  until  the  middle  wire  points  upon  it,  and 
then,  by  means  of  the  "  Azimuth  Screw,"  which  gives  a  slight  horizontal 
motion  to  one  of  the  Y's,  follow  the  star  until  the  indicated  moment  of  its  tran- 
sit ;  i.e.,  until  the  clock  (corrected  for  clock  error)  shows  on  its  face  the  star's 
right  ascension.  If  the  clock  correction  had  been  known  with  absolute  exact- 
ness, the  instrument  would  now  be  truly  in  the  meridian :  as  the  clock  error, 
however,  is  only  approximate,  the  instrument  will  only  be  approximately  in 
the  meridian ;  but  —  and  this  is  the  essential  point  —  it  will  be  very  much 
more  nearly  so  than  at  the  beginning  of  the  operation.  The  supposed  incor- 
rectness, amounting  perhaps  to  one  or  two  seconds,  in  the  time  at  which  the 
instrument  was  set  on  the  circumpolar  star  will,  on  account  of  the  slew  mo- 
tion of  the  star,  make  almost  no  perceptible  difference  in  the  direction  given 
to  the  axis. 

A  repetition  of  the  operation  may  possibly  be  needed  to  secure  all  the 
desired  precision.  The  accuracy  of  this  azimuth  adjustment  can  then  be 
verified  by  three  successive  "  culminations  "  or  transits  of  the  pole  star,  or 
any  other  circumpolar.  The  interval  occupied  in  passing  from  the  upper  to 
the  lower  culmination  on  the  west  side  of  the  meridian  ought,  of  course 
to  be  exactly  equal  to  the  time  on  the  eastern  side ;  i.e.,  twelve  sidereal 
hours. 


ASTRONOMICAL   INSTRUMENTS. 


61.  The  final  test  of  all  the  adjustments,  and  of  the  accurate  going 
of  the  clock,  is  obtained  by  observing  a  number  of  Almanac  stars  of 
widely  different  declination.     If  they  all  indicate  identically  the  same 
clock  correction,  the  instrument  is  in  adjustment ;  if  not,  and  if  the 
differences  are  not  very  great,  it  is  possible   to   deduce   from   the 
observations  themselves  the  true  clock  error,   and  the  adjustment 
errors  of  the  instrument. 

It  is  to  be  added,  in  this  connection,  that  the  astronomer  can  never  assume 
that  adjustments  are  perfect :  even  if  once  perfect,  they  would  not  stay  so,  on 
account  of  changes  of  temperature  and  other  causes.  Nor  are  observations 
ever  absolutely  accurate.  The  problem  is,  from  observations  more  or  less 
inaccurate  but  honest,  with  instruments  more  or  less  maladjusted  but  firm,  to 
find  the  result  that  would  have  been  obtained  by  a  perfect  observation  with 
a  perfect  and  perfectly  adjusted  instrument.  It  can  be  more  nearly  done 
than  one  might  suppose.  But  the  discussion  of  the  subject  belongs  to 
Practical  Astronomy,  and  cannot  be  entered  into  here. 

62.  Prime  Vertical  Instrument.— For  certain  purposes,  a  Transit 
Instrument,  provided  with  an  apparatus  for  rapid  reversal,  is  turned 
quarter-way  round  and  mounted  with  the  axis  north  and  south,  so 
that  the  plane  of  rotation  lies  east  and  west,  instead  of  in  the  meri- 
dian.    It  is  then  called  a  Prime  Vertical  Transit. 

63.  The  Meridian  Circle.  —  In 
order  to  determine  the  Declina- 
tion or  Polar  Distance  of  an 
object,  it  is  necessary  to  have 
some  instrument  for  measuring 
angles ;  mere  time-observations 
will  not  suffice.  The  instrument 
most  used  for  this  purpose  is  the 
Meridian  Circle,  or  Transit  Cir- 
cle, which  is  simply  a  transit  in- 
strument, with  a  graduated  circle 
attached  to  its  axis,  and  revolv- 
ing with  the  telescope.  Some- 
times there  are  two  circles,  one 

FIG.  19. -The  Meridian  Circle  (Schematic).        at  Gach  6nd  °f  the  axis' 

Fig.  19  represents  the  instru- 
ment "  schematically,"  showing  merely  the  essential  parts.  Fig.  20 
is  a  meridian  circle,  with  a  4-inch  telescope,  constructed  by  Fauth 
&Co. 


ASTJRONOMICAL   INSTRUMENTS 


43 


FIG.  20.  —  Meridian  Circle. 


A,  B,  C,  D,  the  Beading  Microscopes. 
K,  the  Graduated  Circle. 
H,  the  Roughly  Graduated  Setting  Circle. 
/,  the  Index  Microscope.     This  is  usually,  however, 
placed  halfway  between  A  and  I). 


F,  the  Clamp.     G,  the  Tangent  Screw. 
LL,  the  Level,  placed  in  position  only  occasionally 
M,  the  Eight  Ascension  Micrometer. 
WW,  Counterpoises,  which  take  part  of  the  weight 
of  the  instrument  off  from  the  Y's. 


44 


ASTRONOMICAL   INSTRUMENTS. 


In  observatory  instruments  the  circle  is  usually  from  two  to  four  feet  in 
diameter  ;  larger  circles  were  once  used,  but  it  is  found  that  their  weight, 
and  the  consequent  strains  and  flexures,  render  them  actually  less  accurate 
than  the  smaller  ones.  The  utmost  resources  of  mechanical  art  are  ex- 
hausted in  making  the  graduation  as  precise  as  possible  and  in  providing  for 
its  accurate  reading,  as  well  as  in  securing  the  maximum  firmness  and  sta- 
bility of  every  part  of  the  instrument.  The  actual  divisions  are  usually 
5'  apart  (in  very  large  instruments  sometimes  only  2'),  but  the  circle  is 
"  read  "  to  seconds  and  tenths  of  seconds  of  arc  by  means  of  reading  micro- 
scopes, from  two  to  six  in  number,  fixed  to  the  pier  of  the  instrument.  In  a 
circle  of  forty  inches  diameter,  1"  is  a  little  less  than  y^^  °f  an  inch, 
inch),  so  that  the  necessity  of  fine  workmanship  is  obvious. 


64.  The  Reading  Microscope  (Fig.  21).  —  This  consists  essen- 
tially of  a  compound  microscope,  which  forms  a  magnified  image  of 
the  graduation  at  the  focus  of  its  object-glass,  where  this  image  is 

viewed  by  a  positive  eye-piece.  At  the 
place  where  the  image  is  formed  a  pair  of 
O  parallel  spider-lines  or  a  cross  is  placed, 
movable  in  the  plane  of  the  image  by  a 
"micrometer  screw  ";  i.e.,  a  fine  screw 
with  a  graduated  head,  usually  divided  into 
sixty  parts.  One  revolution  of  the  screw 
carries  the  wire  1'  of  arc,  which  makes 
one  division  of  the  screw-head  1",  the 
tenths  of  seconds  being  estimated. 


Limb  of  Circle 


The  adjustment  of  the  microscope  for 
"  runs,"  as  it  is  called  (that  is,  to  make  one 
revolution  of  the  micrometer  screw  exactly 
equal  to  1'),  is  effected  as  follows.  By  setting 
the  wires  first  on  one  of  the  graduation  marks 
visible  in  the  field  of  view,  and  then  on  the 
PIG.  21. —The  Reading  Microscope,  next  mark,  it  is  immediately  evident  whether 

five  revolutions  of  the  screw  "run"  over  or 

fall  short  of  5'  of  the  graduation.  If  they  overrun,  it  shows  that  the  image 
of  the  graduation  formed  by  the  microscope  objective  is  too  small  to  fit  the 
screw,  and  vice-versa.  Now,  by  simply  increasing  or  decreasing  the  distance 
AB  between  the  objective  and  the  micrometer  box,  the  size  of  the  image 
can  be  altered  at  will,  and  the  objective  is  therefore  so  mounted  that  this 
can  be  done.  Of  course,  every  change  in  the  length  of  the  microscope  tube 
will  also  require  a  readjustment  of  the  distance  between  the  "limb,"  or 
graduated  surface,  of  the  circle  and  the  microscope,  in  order  to  secure  distinct 
vision ;  but  by  a  few  trials  the  adjustment  is  easily  made  sufficiently  precise 


ASTRONOMICAL   INSTRUMENTS. 


45 


The  reading  of  the  circle  is  as  follows  :  An  extra  index-microscope, 
with  low  power  and  large  field  of  view,  shows  l\v  inspection  the  de- 
grees and  minutes.  The  reading-microscopes  are  only  used  to  give 
the  odd  seconds,  which  is  done  by  turning  the  screw  until  the  parallel 
spider-lines  are  made  to  include  one  of  the  graduation  lines  half-way 
between  themselves  ;  the  head  of  the  screw  then  shows  directly  the 
seconds  and  tenths,  to  be  added  to  the  degrees  and  minutes  shown 
by  the  index.  Thus  in  Fig.  22,  the  reading  of  the  microscope  is 
3'  22".l,  the  3'  being  given  by  the  scale  in  the  field,  the  22".l  by  the 
screw-head. 


FIG.  22.  — Field  of  View  of  Reading  Microscope. 


65.  Method  of  observing  a  Star. — A  minute  or  two  before  the  star 
reaches  the  meridian  the  instrument  is  approximately  pointed,  so  that 
the  star  will  come  into  the  field  of  view.     As  soon  as  it  makes  its 
appearance,  the  instrument 

is  moved  by  the  slow-mo- 
tion tangent -screw  until 
the  star  is  "bisected"  by 
the  fixed  horizontal  wire 
of  the  reticle,  and  the 
star  is  kept  bisected  until 
it  reaches  the  middle  ver- 
tical wire  which  marks  the 
meridian.  The  microscopes  are  then  read,  and  their  mean  result  is 
the  star's  "  circle-reading." 

Frequently  the  star  is  bisected,  not  by  moving  the  whole  instrument,  but 
by  means  of  a  "  micrometer  wire,"  which  moves  up  and  down  in  the  field  of 
view.  The  micrometer  reading  then  has  to  be  combined  with  the  reading 
of  the  microscope,  to  get  the  true  circle-reading. 

66.  Zero  Points. — In   determining   the    declination   or  meridian 
altitude  of  a  star  by  means  of  its  circle-reading,  it  is  necessary  to 
know  the  "zero  point"  of  the  circle.     For  declinations,  the  "zero 
point"  is  either  the  polar  or  the  equatorial  reading  of  the  circle ;  i.e., 
the  reading  of  the  circle  when  the  telescope  is  pointed  at  the  pole 
or  at  the  equator. 

The  "  polar  point"  may  be  found  by  observing  some  circumpolar 
star  above  the  pole,  and  again,  twelve  hours  later,  below  it.  When 
the  two  circle-readings  have  been  duly  corrected  for  refraction  and 
instrumental  errors,  their  mean  will  be  the  polar  point. 


46  ASTRONOMICAL   INSTRUMENTS. 

Suppose,  for  instance,  that  5  Ursse  Minoris,  at  the  "  upper  culmination," 
gives  a  corrected  reading  of  52°  18'  25".3,  while  at  the  lower  culmination  the 
reading  is  45°  31'  35".7,  then  the  mean  of  these,  48°  55'  00".5,  is  the  polar 
point,  and  of  course  the  equatorial  reading  is  138°  55'  00".5,  —  just  90C 
greater.  The  polar  distance  of  the  star  would  be  the  half-difference  of  the 
two  readings,  or  3°  23'  24".8. 

67.  Nadir  Point.  —  The  determination  of  the  polar  point  requires 
two  observations  of  the  same  star  at  an  interval  of  twelve  hours.  It  is 
often  difficult  to  obtain  such  a  pair ;  moreover,  the  refraction  compli- 
cates the  matter,  and  renders  the  result  less  trustworthy.  Accord- 
ingly it  is  now  usual  to  use  the  nadir  or  the  horizontal  reading  as  the 
zero,  rather  than  the  polar  point. 

The  nadir  point  is  determined  by  pointing  the  telescope  down- 
wards to  a  basin  of  mercury,  moving  the  telescope  until  the  image 
of  the  horizontal  wire  of  the  reticle,  as  seen  by  reflection,  coincides 
with  the  wire  itself.  Since  the  reticle  is  exactly  in  the  principal 
focus  of  the  object-glass,  rays  of  light  emitted  by  any  point  in  the 
reticle  will  become  a  parallel  beam  after  passing  the  lens,  and  if  this 
beam  strikes  a  plane  mirror  perpendicularly  and 
is  returned,  the  rays  will  come  just  as  if  from  a 
real  object  in  the  sky,  and  will  form  an  image 
at  the  focal  plane.  When,  therefore,  the  image 
of  the  central  wire  of  the  reticle,  seen  in  the 
mercury  basin  by  reflection,  coincides  with  the 
^Reticle  WIYQ  itself,  we  know  that  the  line  of  collimation 

FIG.  23.  must  be  exactly  perpendicular  to  the  surface  of 

The  Commating  Eye-Piece.  the  mercury  ;  i.e. ,  vertical. 

To  make  the  image  visible  it  is  necessary  to  illuminate  the  reticle  by  light 
thrown  towards  the  object-glass  from  behind  the  wires,  instead  of  light 
coming  from  the  object-glass  towards  the  eye  as  usual.  This  peculiar  illu- 
mination is  commonly  effected  by  means  of  Bohnenberger's  "collimating 
eye-piece,"  shown  in  Fig.  23.  In  the  simplest  form  it  is  merely  a  common 
Ramsden  eye-piece,  with  a  hole  in  one  side,  and  a  thin  glass  plate  inserted 
at  an  angle  of  45°.  A  light  from  one  side,  entering  through  the  hole,  will  be 
(partially)  reflected  towards  the  wires,  and  will  illuminate  them  sufficiently. 

The  horizontal  point  of  course  differs  just  90°  from  the  nadir  point.  It 
may  also  be  found  independently  by  noting  the  circle-readings  of  some  star 
observed  one  night  directly,  and  the  next  night  by  reflection  in  mercury ;  or, 
if  the  star  is  a  close  circumpolar,  both  observations  may  be  made  the  same 
evening,  one  a  few  minutes  before  its  meridian  passage,  the  other  just  as 
long  after.  But  the  method  of  the  collimating  eye-piece  is  fully  as  accurate 
and  vastly  more  convenient. 


ASTRONOMICAL   INSTRUMENTS.  47 

68.  Differential  Use  of  the  Instrument. — We  now  know  the  places  of 
several  hundred  stars  with  so  much  precision  that  in  many  cases  it  is  quite 
sufficient  to  observe  one  or  two  of  these  "  standard  stars  "  in  connection  with 
the  bodies  whose  places  we  wish  to  determine.     The  difference  between  the 
declination  of  the  known  star  and  that  of  any  star  whose  place  is  to  be 
determined,  will,  of  course,  be  simply  the  difference  of  their  circle-readings, 
corrected  for  refraction,  etc.     The  meridian  circle  is  said  to  be  used  "differ- 
entially "  when  thus  treated. 

69.  Errors  of  Graduation,  etc.  —  If  the  circle  is  from  a  reputable 
maker,  and  has  four  or  six  microscopes,  and  if  the  observations  are 
carefully  made  and  all  the  microscopes  read  each  time,   results  of 
sufficient  precision  for  most  purposes  may  be  obtained  by  merely 
correcting  the  observations   for  "runs"  and  refraction.     The    out- 
standing errors  ought  not  to  exceed  a  second  or  two.     But  when  the 
tenths  of  a  second  are  in  question,  the  case  is  different.     It  will  not 
then  do  for  the  astronomer  to  assume  the  accuracy  of  the  graduation 
of  his  circle,  but  he  must  investigate  the  errors  of  its  divisions,  the 
errors  of  the  micrometer  screws  in  the  microscopes,  the  flexure  of  the 
telescope,  and   the   effect   of   differences  of  temperature   in  shifting 
the  zero  points  of  the  circle,  by  slightly  disturbing  the  position  or 
direction  of  the  microscopes.     Of  course  this  is   not  the   place  to 
enter  into  such  details,  but  it  is  an  opportunity  to  impress  again  upon 
the  student  the  fact  that  truth  and  accuracy  are  only  attainable  by 
immense  painstaking  and  labor. 

70.  Mural  Circle.  —  This  instrument  is  in  principle  the  same  as  the 
meridian  circle,  which  has  superseded  it.     It  consists  of  a  circle,  carrying  a 
telescope  mounted  on  the  face  of  a  wall  of  masonry  (as  its  name  implies) 
and  free  to  revolve  in  the  plane  of  the  meridian.     The  wall  furnishes  a  con- 
venient support  for  the  microscopes. 

71.  Altitude  and  Azimuth,  or  Universal,  Instrument.  —  Since  the 
transit  instrument  and  meridian  circle  are  confined  to  the  plane  of 
the  meridian,  their  usefulness  is  obviously  limited.     Meridian  ob- 
servations are  better  and  more  easily  used  than  any  others,  but  are 
not  always  attainable.     We  must  therefore  have  instruments  which 
will  follow  an  object  to  any  part  of  the  heavens. 

The  altitude  and  azimuth  instrument  is  simply  a  surveyor's  theodo- 
lite on  a  large  scale.  It  has  a  horizontal  circle  turning  upon  a  verti- 
cal axis,  and  read  by  verniers  or  microscopes.  Upon  this  circle,  and 
turning  with  it,  are  supports  which  carry  the  horizontal  axis  of  the 
telescope  with  its  vertical  circle,  also  read  by  microscopes.  Obvi- 


48  ASTRONOMICAL   INSTRUMENTS. 

ously  the  readings  of  these  two  circles,  when  the  instrument  is  prop- 
erly adjusted  and  the  zero  points  determined,  will  give  the  altitude 


FIG.  24.  —  Altitude  and  Azimuth  Instrument. 


and  azimuth  of  the  body  pointed  on.     Fig.  24  represents  a  small  ir 
strument  of  this  kind. 


ASTRONOMICAL   INSTRUMENTS. 


49 


72.  The  Equatorial. — -The  essential  characteristic  of  this  instrument 
is  that  its  principal  axis,  i.e.,  the  axis  which  rests  in  fixed  bearings, 
instead  of  being  either  horizontal  or  vertical,  is  inclined  at  an  angle 
equal  to  the  latitude  of  the  place,  and  directed  towards  the  pole,  thus 
placing  it  parallel  to  the  earth's  axis  of  rotation.  This  axis  of  the 
instrument  is  called  its  polar  axis;  and  the  graduated  circle  which  it 
carries,  and  which  is  parallel  to  the  celes- 
tial equator,  is  called  the  hour-circle,  be- 
cause its  reading  gives  the  hour-angle  of 
the  body  upon  which  the  telescope  hap- 
pens to  be  pointed.  Sometimes,  also,  it  is 
called  the  Right  Ascension  Circle.  Upon 
this  polar-axis  are  secured  the  bearings 
of  the  declination  axis,  which  is  perpen- 
dicular to  the  polar  axis,  and  carries  the 
telescope  itself  and  the  declination  circle. 

In  the  instruments  before  described,  the 
telescope  is  a  mere  pointer,  and  wholly 
subsidiary  to  the  circles  ;  in  the  equatorial 
the  telescope  is  usually  the  main  thing, 
and  the  circles  are  subordinate,  serving 
only  to  aid  the  observer  in  finding  or 
identifying  the  body  upon  which  the  telescope  is  directed. 

Fig.  25  exhibits  schematically  the  ordinary  form  of  equatorial 
mounting,  of  which  there  are  numerous  modifications.  Fig  26  is  the 
23-inch  Clark  telescope  at  Princeton,  and  Fig.  27  is  the  4-foot 
Melbourne  reflector.  The  frontispiece  is  the  great  Lick  telescope 
of  thirty-six  inches  diameter. 

The  advantages  of  the  equatorial  mounting  for  a  large  telescope 
are  very  great  as  regards  convenience.  In  the  first  place,  when  the 
telescope  is  once  pointed  upon  a  star  or  planet,  it  is  only  necessary 
to  turn  the  polar  axis  with  a  uniform  motion  in  order  to  "  follow  "  the 
star,  which  otherwise  would  be  carried  out  of  the  field  of  view  in  a 
few  moments  by  the  diurnal  motion.  This  motion,  since  it  is  uni- 
form, can  be,  and  in  all  large  instruments  usually  is,  given  by  clock- 
work, with  a  continuous  regulator  of  some  kind,  similar  to  that  used 
in  the  chronograph.  The  instrument  once  directed  and  clamped, 
and  the  clock-work  started,  the  object  will  continue  apparently  im- 
movable in  the  field  of  view  as  long  as  may  be  desired. 

In  the  next  place,  it  is  very  easy  to  find  an  object,  even  if  invisible 
to  the  naked  eye,  like  a  faint  comet  or  nebula,  or  a  star  in  the  day- 


FIG.  25. 
The  Equatorial  (Schematic). 


50 


ASTRONOMICAL   INSTRUMENTS. 


time,  provided  we  know  its  declination  and  right  ascension,  and 
have  the  sidereal  time  ;  for  which  reason  a  sidereal  clock  or  chro- 
nometer is  an  indispensable  adjunct  of  the  equatorial. 


FIG.  26.  —  The  23-inch  Princeton  Telescope. 


To  find  an  object,  the  telescope  is  turned  in  decimation  until  the  reading 
of  the  declination  circle  corresponds  to  the  declination  of  the  object,  and 
then  the  polar  axis  is  turned  until  the  hour-circle  of  the  instrument  (not  to 
be  confounded  with  an  hour-circle  in  the  sky)  reads  the  hour-angle  of  the 
object.  This  hour-angle,  it  will  be  remembered,  is  simply  the  difference  be- 


ASTRONOMICAL   INSTRUMENTS. 


51 


tween  the  sidereal  time  and  the  right  ascension  of  the  object.  The  hour- 
angle  is  east  if  the  right  ascension  exceeds  the  time;  west,  if  it  is  less. 
When  the  telescope  is  thus  set,  the  object  will  be  found  (with  a  low  mag- 
nifying power)  in  the  field  of  view,  unless  it  is  near  the  horizon,  in  which 
case  refraction  must  be  taken  into  account. 


FIG.  27.  — The  Melbourne  Reflector. 

While  the  instrument  cannot  give  very  accurate  determinations  of 
the  positions  of  bodies  by  the  direct  readings  of  its  circles,  on  account 
of  the  irregular  flexures  of  its  axes,  it  may  do  so  indirectly ;  that  is, 
it  ma}^  be  used  to  determine  very  accurately  the  difference  between 
the  right  ascension  and  declination  of  a  comet  or  planet,  for  instance, 
and  that  of  some  neighboring  star,  whose  place  has  been  already 
determined  by  the  meridian  circle ;  and  this  is  one  of  the  most  im- 
portant uses  of  the  instrument. 


52 


ASTRONOMICAL   INSTRUMENTS. 


73.     The  Micrometer. — Micrometers  of  various  sorts  are  employed 
for  the  purpose.     The  most  common  and  most  generally  useful  is  the 
so-called  "filar  position-micrometer"  Fig.  28,  which  is  an  indispen- 
sable   auxiliary   of    every    good 
telescope. 

It  is  a  small  instrument,  much 
like  the  upper  part  of  the  read- 
ing microscope,  but  more  com- 
plicated. It  usually  contains  a 
reticle  of  fixed  wires,  two  or 
three  parallel  to  each  other,  and 
crossed  at  right  angles  by  a 
second  set.  Then  there  are  two 
or  three  wires  parallel  to  the  first 
set,  and  movable  by  an  accu- 
rately made  screw  with  a  gradu- 
ated head  and  a  counter,  or 
scale,  for  indicating  the  number 
of  entire  revolutions  made  by 
the  screw.  The  box  containing 
these  wires,  and  carrying  the  eye-piece  and  screw,  can  itself  be 
turned  around  in  a  plane  perpendicular  to  the  optical  axis  of  the 
telescope,  and  set  in  any  desired  position  ;  for  example,  so  that  the 
movable  wires  shall  be  parallel  to  the  celestial  equator,  while  the 


FIG.  28.  —  The  Filar  Position-Micrometer. 


FIG.  29.  —  Construction  of  the  Micrometer. 


other  set  run  north  and  south.  This  "  position  angle"  is  read  on  a 
graduated  circle,  which  forms  part  of  the  instrument.  Means  of 
illumination  are  provided,  giving  at  pleasure  either  dark  wires  in  a 
bright  field,  or  vice  versa. 

With  this  instrument  one  can  measure  the  distance  (in  seconds  of 
arc),  and  the  direction  between  any  two  stars  which  are  near  enough 
bo  be  seen  at  once  in  the  same  field  of  view.  This  range  in  small 


ASTRONOMICAL   INSTRUMENTS. 


53 


telescopes  may  reach  30'  of  arc  ;  while  in  the  larger  instruments, 
which,  with  the  same  eye-pieces  have  much  higher  magnifying  pow- 
ers, it  is  necessarily  less,  —  not  more  than  from  5'  to  10'. 

74,  A  new  form  of  equatorial,  known  as  the  Equatorial  Coude,  or  Elbowed 
Equatorial,  has  been  recently  introduced  at  the  Paris  Observatory.  With 
large  instruments  of  the  ordinary  form  a  great  deal  of  inconvenience  is  en- 
countered by  the  observer,  in  moving  about  to  follow  the  eye-piece  into  the 
various  positions  into  which  it  is  forced  by  the  inconsiderateness  of  the 


FIG.  30.  —  The  Equatorial  Coude. 

heavenly  bodies.  Moreover,  the  revolving  dome,  which  is  usually  erected  to 
shelter  a  great  telescope,  is  an  exceedingly  cumbrous  and  expensive  affair. 

In  the  Equatorial  Coude",  Fig.  30,  these  difficulties  are  overcome  by  the 
use  of  mirrors.  The  observer  sits  always  in  one  fixed  position,  looking 
obliquely  down  through  the  polar  axis,  which  is  also  the  telescope  tube.1 

The  instrument  (figured  above)  had  an  aperture  of  about  ten  inches,  and 
proved  so  satisfactory  that  in  1891  >a  much  larger  one  with  a  twenty-four- 
inch  lens  was  also  mounted,  and  both  are  now  in  constant  use. 

75.  All  instruments  so  far  described,  except  the  chronometer, 
are  fixed  instruments ;  of  use  only  when  they  can  be  set  up  firmly 
and  carefully  adjusted  to  established  positions.  Not  one  of  them 
would  be  of  the  slightest  use  on  shipboard. 

1  For  description  of  the  siderostat  and  ccelostat  see  Addendum  A. 


54 


ASTRONOMICAL   INSTRUMENTS. 


We  have  now  to  describe  the  instrument  which,  with  the  help  of 
the  chronometer,  is  the  main  dependence  of  the  mariner.  It  is  an 
instrument  with  which  the  observer  measures  the  angular  distance 
between  two  objects  ;  as,  for  instance,  the  sun  and  the  visible  horizon, 
not  by  pointing  first  on  one  and  then  afterwards  on  the  other,  but  by 
sighting  them  both,  simultaneously  and  in  apparent  coincidence;  which 
can  be  done  even  when  he  has  no  fixed  position  or  stable  footing. 

76.  The  Sextant.  — The  graduated  limb  of  the  sextant  is  carried 
by  a  light  framework,  usually  of  metal,  provided  with  a  suitable  handle 
X.  The  arc  is  about  one-sixth  of  a  circle,  as  the  name  implies,  and 


M 


FIG.  31.  — The  Sextaut. 


is  usually  from  five  to  eight  inches  radius.  It  bears  a  graduation  of 
half -degrees,  numbered  as  whole  degrees,  so  that  it  can  measure  any 
angle  less  than  120°. 

An  u  index-arm,"  MNin  the  figure,  is  pivoted  at  the  centre  of  the 
arc,  and  carries  a  vernier  which  slides  along  the  limb,  and  can  be 
fixed  at  any  point  by  a  clamp  and  delicately  moved  by  the  attached 
tangent  screw,  T.  The  reading  of  this  vernier  gives  the  angle 
measured  by  the  instrument.  The  best  instruments  read  to  10". 

Just  over  the  centre  of  motion,  the  "index-mirror"  M,  about 
two  inches  by  one  and  one-half  in  size,  is  fastened  securely  to  the 
index-arm,  so  as  to  be  perpendicular  to  the  plane  of  the  limb.  At 


ASTRONOMICAL   INSTRUMENTS.  55 

H,  the  "  horizon-glass,"  about  an  inch  wide  and  of  the  same  height 
as  the  index-glass,  is  secured  firmly  to  the  frame  of  the  instrument, 
in  such  position  that,  when  the  vernier  of  the  index-arm  reads  zero, 
the  index-mirror  and  horizon-glass  will  be  parallel  to  each  other. 
Only  half  of  the  horizon-glass  is  silvered,  the  upper  half  being  left 
transparent.  E  is  a  small  telescope. 

If  the  vernier  stands  near,  but  not  at  zero,  the  observer  look- 
ing into  the  telescope  will  see  together  in  the  field  of  view  two  sepa- 
rate images  of  the  object ;  and  if,  while  still  looking,  he  slides 
the  vernier  a  little,  he  will  see  that  one  of  the  images  remains  fixed, 
while  the  other  moves.  The  fixed  image  is  due  to  the  rays  which 
reach  the  object-glass  of  the  telescope  directly,  coming  through 
the  unsilvered  half  of  the  horizon-glass  :  the  movable  image,  on  the 
other  hand,  is  produced  by  rays  which  have  suffered  two  reflections, 
—  first,  from  the  index-mirror  to  the  horizon-glass  ;  and  second,  at 
the  lower  half  of  the  horizon-glass.  When  the  two  mirrors  are 
parallel,  and  the  vernier  reads  zero,  the  two  images  coincide,  pro- 
vided the  object  is  at  a  considerable  distance. 

If  now  the  vernier  does  not  stand  at  or  near  zero,  the  observer, 
looking  at  any  object  directly  through  the  horizon-glass,  will  see, 
not  only  that  object,  but  also  whatever  other  object  is  so  situated 
as  to  send  its  rays  to  the  telescope  by  reflection  upon  the  mir- 
rors ;  and  the  reading  of  the  vernier  will  give  the  angle  at  the  instru- 
ment between  the  two  objects  whose  images  thus  coincide;  the  angle 
between  the  planes  of  the  two  mirrors  being  just  half  that  between 
the  objects,  and  the  half -degrees  on  the  limb  being  numbered  as 
whole  ones. 

77.  The  principal  use  of  the  instrument  is  in  measuring  the  altitude 
of  the  sun.  At  sea  the  observer,  holding  the  instrument  with  his  right 
hand  and  keeping  the  plane  of  the  arc  vertical,  looks  directly  towards 
the  visible  horizon  at  the  point  under  the  sun,  through  the  horizon- 
glass  (whence  its  name)  ;  then  by  moving  the  vernier  with  his 
left  hand,  he  inclines  the  index-glass  upwards  until  one  edge  of  the 
reflected  image  of  the  sun  is  brought  just  to  touch  the  horizon-line, 
noting  the  exact  time  by  the  chronometer,  if  necessary.  The  reading 
of  the  vernier,  after  correcting  for  the  semi-diameter  of  the  sun,  the 
dip  of  the  horizon,  the  refraction,  and  the  parallax  (and  for  the 
"  index  -error "  of  the  sextant,  if  the  vernier  does  not  read  strictly 
zero  when  the  mirrors  are  parallel)  gives  the  sun's  true  altitude  at  the 
moment. 


56 


ASTRONOMICAL   INSTRUMENTS. 


78.  On  land  the  visible  horizon  is  of  no  use,  and  we  have  recourse  to  an 
"artificial  horizon"  as  it  is  called.  This  is  merely  a  shallow  basin  of  mercury, 
covered,  when  necessary  to  protect  it  from  the  wind,  with  a  roof  made  of 
glass  plates  having  their  sides  plane  and  parallel. 

In  this  case  we  measure  the  angle  between  the  sun's  image  reflected  in  the 
mercury  and  the  sun  itself.  The  reading  of  the  instrument,  corrected  for 
index-error,  gives  twice  the  sun's  apparent  altitude;  which  apparent  altitude, 

corrected  as  before  for  refraction  and 
parallax,  but  not  for  dip  of  the  horizon, 
gives  the  true  altitude.  The  skilful  use 
of  the  sextant  requires  steadiness  of 
hand  and  considerable  dexterity,  and 
from  the  small  size  of  the  telescope  the 
angles  measured  are  of  course  less  pre- 
cise than  if  determined  by  large  fixed 
instruments.  But  its  portability  and 
applicability  at  sea  render  it  absolutely 
invaluable. 

79.     The  principle  that  the  true  angle 
between  the  objects  whose  images  coin- 
cide is  twice  the  angle  between  the  mir- 
rors (or  between  their  normals)  is  easily 
demonstrated  as  follows  (Fig.  32)  :  — 

The  ray  SM  coming  from  an  object,  after  reflection  first  at  M  (the  index- 
mirror),  and  then  at  H  (the  horizon-glass),  is  made  to  coincide  with  the 
ray  OH  coming  from  the  horizon.  We  must  prove  that  the  angle  SEO,  be- 
tween the  object  and  the  horizon,  as  seen  from  the  point  E  in  the  instrument, 
is  double  the  angle  Q,  between  MQ  and  HQ,  which  are  normals  to  the  mir- 
rors, and  therefore  double  Q',  which  is  the  angle  between  the  planes  of  the 
mirrors. 

First,  from  the  law  of  reflection,  we  have, 

SMP=HMP,  or   SMH=2xPMH. 


PIG.  32.  —  Principle  of  the  Sextant. 


Similarly, 

From  the  geometric  principle  that  the  exterior  angle  SMH  of  the  triangle 
HME  is  equal  to  the  sum  of  the  opposite  interior  angles  at  H  and  E,  we  get 

HEM=  SMH-MHE  =  2  PMH-2  MHQ  =  2(PMH-MHQ) . 
Similarly,  from  the  triangle  HMQ,  we  have 

H  QM  =  PMH  -  MHQ, 
which  is  half  the  value  just  found  for  HEM,  and  proves  the  proposition 


ASTRONOMICAL   INSTRUMENTS.  57 

Of  course  with  the  sextant,  as  with  all  other  instruments,  it  is 
necessary  for  the  observer  who  aims  at  the  utmost  precision  to  in- 
vestigate, and  take  into  account  its  errors  of  graduation,  construction 
and  adjustment  ;  but  their  discussion  lies  beyond  our  scope. 

80,  Besides  the  instruments  we  have  described,  there  are  many 
others  designed  for  special  work,  some  of  which,  as  the  zenith  tele- 
scope and  heliometer,  will  be  mentioned  hereafter  as  it  becomes 
necessary.  There  is  also  a  whole  class  of  physical  instruments, 
photometers,  spectroscopes,  heat-measuring  appliances,  and  photo- 
graphic apparatus,  which  will  have  to  be  considered  in  due  time. 

But  with  clock,  meridian  circle,  and  equatorial  and  their  usual 
accessories,  all  the  fundamental  observations  of  theoretical  and 
spherical  astronomy  can  be  supplied.  The  chronometer  and  sextant 
are  practically  the  only  astronomical  instruments  of  any  use  at  sea. 


58 


DIP  OF  THE  HORIZON. 


CHAPTER  III. 


CORRECTIONS  TO  ASTRONOMICAL  OBSERVATIONS,  DIP  OF  THE 
HORIZON,  PARALLAX,  SEMI-DIAMETER,  REFRACTION,  AND 
TWILIGHT. 

81.  Dip  of  the  Horizon.  —  In  observations  of  the  altitude  of 
a  heavenly  body  at  sea,  where  the  measurement  is  made  from 
the  sea-line,  a  correction  is  needed  on  account  of  the  fact  that 
this  visible  horizon  does  not  coincide  with  the  true  astronomical 
horizon  (which  is  90°  from  the  zenith)  ,  but 
falls  sensibly  below  it  by  an  amount  known 
as  the  Dip  of  the  Horizon.  The  amount  of 
this  dip  depends  upon  the  size  of  the  earth 
and  the  height  of  the  observer's  eye  above 
the  sea-level. 

In  Fig.  33,  G  is  the  centre  of  the  earth, 
AB  a  portion  of  its  level  surface,  and  0  the 
observer,  at  an  elevation  h  above  A.  The 
line  OH  is  truly  horizontal,  while  the  tangent 

Fie.  33.  -Dip  of  the  Horizon,    line,   OB,   corresponds    to    the    line    drawn 
from   the  eye  to    the  visible  horizon.     The 

angle  HOB  is  the  dip.     This  is  obviously  equal  to  the  angle  OCB 
at  the  centre  of  the  earth,  if  we  regard  the  earth  as  spherical,  as  we 
may  do  with  quite  sufficient  accuracy  for  the  purpose  in  hand. 
From  the  right-angled  triangle  OBO  we  have  directly 


cos  OCB  = 


BG 
CO 


Putting  R  for  the  radius  of  the  earth,  and  A  for  the  dip,  this  becomes 


cos  A  = 


R 
R  +  h' 


This  formula  is  exact,  but  inconvenient,  because  it  gives  the  small  angle 
A  by  means  of  its  cosine.  Since,  however,  1  —  cos  A  =  2  sin2 £  A,  we  easily 
obtain  the  following :  — 

h 


DIP   OF   THE   HORIZON.  59 

This  gives  the  true  depression  of  the  sea  horizon,  as  it  would  be  if  the 
line  of  sight,  drawn  from  the  eye  to  the  horizon  line,  were  straight.  On 
account  of  refraction  it  is  not  straight,  however,  and  the  amount  of  this 
"  terrestrial  refraction  "  is  very  variable  and  uncertain.  It  is  usual  to 
diminish  the  dip  computed  from  the  formula  by  one-eighth  its  whole  amount. 

An  approximate  formula1  for  the  dip  is 

A  (in  minutes  of  arc)  =  V/i  (feet)  ; 

or,  in  words,  the  square  root  of  the  elevation  of  the  eye  (in  feet)  gives 
the  dip  in  minutes.     This  gives  a  value  about  -^  part  too  large. 

Since  the  dip  is  applicable  onhr  to  sextant  observations  made  at 
sea,  where,  from  the  nature  of  the  instrument,  and  the  rising  and 
falling  of  the  observer  with  the  vessel's  motion,  it  is  not  possible  to 
measure  altitudes  more  closely  than  within  about  15",  there  is  no 
need  of  any  extreme  precision  in  its  calculation. 

1  This  approximate  formula  may  be  obtained  thus  :  — 


But  since  -  is  a  very  small  fraction,  it  may  be  neglected  in  the  divisor  [  1  +  -  \ 
R  \       RJ 

and  the  expression  becomes  simply, 

»  <  i  - 

=  -;   whence  sin  \  A  =  -J—  . 
R  \2  R 


Since  A  is  a  very  small  angle, 

A  =  sin  A  =  2  sin  ^  A,       so  that 


To  reduce  radians  to  minutes,  we  must  multiply  by  3438,  the  number  of  minutes 
in  a  radian.  (Art.  6,  page  7.)  Accordingly, 

A'  (in  minutes  of  arc)  =  3438     /—  . 

.     \2  R 

If  we  express  h  in  feet,  we  must  also  use  the  same  units  for  E.  The  mean 
radius  of  the  earth  is  about  20,884,000  feet,  one-half  of  which  is  10,442,000,  and 
the  square  root  of  this  is  3231  ;  so  that  the  formula  becomes 

Al     3438 
A==3231 

which  is  near  enough  to  that  given  in  the  text. 

In  fact,  the  refraction  makes  so  much  difference  that  even  after  taking  the 


numerical  factor,  —2?,  as  unity,  the  formula  still  gives  A1  about  ^  part  too  large. 
o231 


The  formula  A'  =  V  3  h  (metres}  is  yet  more  nearly  correct 


60 


CORRECTIONS   TO    OBSERVATIONS. 


82.  Parallax.  —  In  the  most  general  sense, ' 4  parallax  "  is  the  change 
of  a  body's  direction  resulting  from  the  observer's  displacement.  In 
the  restricted  and  technical  sense  in  which  we  are  to  employ  it  now, 
it  may  be  defined  as  the  difference  between  the  direction  of  a  body  as 
actually  observed  and  the  direction  it  would  have  if  seen  from  the  earth's 
centre.  Thus  in  the  figure,  Fig.  34,  where  the  observer  is  supposed 
to  be  at  0,  the  position  of  P  in  the  sky  (as  seen  from  0)  would  be 
marked  b}-  the  point  where  OP  produced  would  pierce  the  celestial 
sphere.  Its  position  as  seen  from  G  would  be  determined  in  the 
same  way  by  producing  CP  to  which  OX  is  drawn  parallel.  The 
angle  POX,  therefore,  or  its  equal,  OPO,  is  the  parallax  of  P  for 
an  observer  at  0. 

Obviously,  from  the  figure,  we  may  also  give  the  following  defini- 
tion of  the  parallax.  It  is  the  angu- 
lar distance  (number  of  seconds  of 
arc)  between  the  observer's  station  and 
the  centre  of  the  earth's  disc,  as  seen 
from  the  body  observed.  The  moon's 
parallax  at  any  moment  for  me  is  my 
angular  distance  from  the  earth's  cen- 
tre, as  seen  by ' '  the  man  in  the  moon." 
When  a  body  is  in  the  zenith  its 
parallax  is  zero,  and  it  is  a  maxi- 
mum at  the  horizon.  In  all  cases  it 
depresses  a  body,  diminishing  the 
altitude  without  changing  the  azimuth. 

The ' '  law  "  of  the  parallax  is,  that  it  varies  as  the  sine  of  the  zenith  dis- 
tance directly,  and  inversely  as  the  linear  distance  (in  miles)  of  the  body. 
This    follows    easily   from    the    triangle    OOP,    where   we    have 
PC:  0(7=  sin  COP:  sin  CPO. 

Put  D  for  PC,  the  distance  of  the  body  from  the  earth ;  R  for 
the  earth's  radius,  CO ;  p  for  OPO,  the  parallax  ;  £  for  ZOP,  the  appar- 
ent zenith  distance,  and  remember  that  the  sine  of  £  is  equal  to  the 
sine  of  its  supplement,  COP:  we  then  have  as  the  translation  of 
the  above  proportion, 

D :  R  =  sin  £ :  sin  p. 


FIG.  34.  —  Diurnal  Parallax. 


This  gives  us 


T> 

siup  —  —  sinf ; 


or,  from  Art.  6,  since  p  is  always  a  small  angle, 
p"  =  206265"-^-  sin  £. 


PARALLAX.  61 

83.  Horizontal  Parallax.  —  When  a  body  is  at  the  horizon  (Ph  in 

the  figure),  then  £  becomes  90°,  and  sin  £  =1.     In  this  case  the  par-          \J 
allax  reaches  its  maximum  value,  which  is  called  the  horizontal  paral- 
lax of  the  body.     Taking^  as  the  symbol  for  this,  we  have 

7?  7? 

sinj)A=—  ;  or,  nearly  enough,  ph  =  206265"  —  . 

Comparing  this  with  the  formula  above,  we  see  that  the  parallax  of 
a  body  at  any  zenith  distance  equals  the  horizontal  parallax  multiplied 
by  the  sine  of  the  zenith  distance;  i.e.,  p  =  ph  sin £. 

N.  B.  A  glance  at  the  figure  will  show  that  we  may  define  the 
horizontal  parallax,  OPkC,  of  any  body,  as  the  angular  semi-diameter 
of  the  earth  seen  from  that  body.  To  say,  for  instance,  that  the  sun's 
horizontal  parallax  is  8". 8,  amounts  to  saying  that,  seen  from  the  sun, 
the  earth's  apparent  diameter  is  twice  8". 8,  or  17". 6. 

84.  Relation  between  Horizontal  Parallax  and  Distance. — Since 
we  have 

sin  ph  =  - , 

it  follows  of  course  that  D  =  R  -5-  smph ; 
or,  (nearly)  D  = 

Ph 

If  the  sun's  parallax  equals  8"-8, 

its  distance  =  206265  x  R  =  23439  R. 

8.8 

85.  Equatorial  ParaUax.  —  Owing  to  the  < '  ellipticity  "  or  "  ob- 
lateness"  of  the   earth   the   horizontal   parallax   of    a  body  varies 
slightly  at  different  places,  being  a  maximum  at  the  equator,  where 
the  distance  of  an  observer  from  the  earth's  centre  is  greatest.     It 
is  agreed  to  take  as  the  standard  the  equatorial  horizontal  parallax ; 
i.e.,  the  earth's  equatorial  semi-diameter  as  seen  from  the  body. 

86.  Diurnal  Parallax.  —  The  parallax  we  have  been  discussing  is 
sometimes  called  the  diurnal  parallax,  because  it  runs  through  all  its 
possible  changes  in  one  day. 

When  the  sun,  for  instance,  is  rising,  its  parallax  is  a  maximum,  and  bv 
throwing  it  down  towards  the  east,  increases  its  apparent  right  ascension. 
At  noon,  when  the  sun  is  on  the  meridian,  its  parallax  is  a  minimum,  and 


62  CORRECTIONS  TO  OBSERVATIONS. 

affects  only  the  declination.  At  sunset  it  is  again  a  maximum,  but  now 
throws  the  sun's  apparent  place  down  towards  the  west.  Although  the  sun 
is  invisible  while  below  the  horizon,  yet  the  parallax,  geometrically  considered, 
again  becomes  a  minimum  at  midnight,  regaining  its  original  value  at  the 
next  sunrise. 

The  qualifier,  "  diurnal,"  is  seldom  used  except  when  it  is  neces- 
sary to  distinguish  between  this  kind  of  parallax  and  the  annual 
parallax  of  the  fixed  stars,  which  is  due  to  the  earth's  orbital  motion. 
The  stars  are  so  far  away  that  they  have  no  sensible  diurnal  parallax 
(the  earth  is  an  infinitesimal  point  as  seen  from  them)  ;  but  some  of 
them  do  have  a  slight  and  measurable  annual  parallax,  by  means 
of  which  we  can  roughly  determine  their  distances.  (Chap.  XIX.) 

87.  Smallness  of  Parallax.  —  The  horizontal  parallax  of  even  the 
nearest  of  the  heavenly  bodies  is  always  small.     In  the  case  of  the 
moon  the  average  value  is  about  57',  varying  with  her  continually 
changing  distance.     Excepting  now  and  then  a  stray  comet,  no  other 
heavenly  body  ever  comes  within  a  distance  a  hundred  times  as  great 
as  hers.     Venus   and   Mars   approach   nearest,  but  the  parallax  of 
neither  of  them  ever  reaches  40". 

88.  Semi-Diameter. — In  order  to  obtain  the  true  altitude  of  an 
object  it  is  necessary,  if  the  edge,  or  "fo'mfr,"  as  it  is  called,  has  been 
observed,  to  add  or  deduct  the  apparent  semi-diameter  of  the  object. 
In  most  cases  this  will  be  sensibly  the  same  in  all  parts  of  the  sky, 
but  the  moon  is  so  near  that  there  is  quite  a  perceptible  difference 
between  her  diameter  when  in  the  zenith  and  in  the  horizon. 


ren 


A  glance  at  Fig.  34  shows  that  in  the  zenith  the  moon's  distance  is  less 
than  at  the  horizon,  by  almost  exactly  the  earth's  radius  —  the  difference 
between  the  lines  OZ  and  OPh.  Now  this  is  very  nearly  one-sixtieth  part 
of  the  moon's  distance,  and  consequently  the  moon,  on  a  night  when  its 
apparent  diameter  at  rising  is  30',  will  be  30"  larger  when  near  the  zenith. 
Since  the  semi-diameter  given  in  the  almanac  is  what  would  be  seen  from  the 
centre  of  the  earth,  every  measure  of  the  moon's  distance  from  stars  or  from 
the  horizon  will  require  us  to  take  into  account  this  "  augmentation  of  the 
semi-diameter,"  as  it  is  technically  called. 

The  formula,  easily  deduced  from  the  figure  by  remembering  that  the 
angle  PCO  =  £—p  (zenith  distance  — parallax),  and  that  the  apparent  and 
"almanac"  diameters  will  be  inversely  proportional  to  the  two  distances 
OP  and  CP,  is 

apparent  semi-diameter  =  almanac  s.  d.  X  — — TZ^ — r- 

sin  (£-p) 


KEFKACTION. 


63 


This  measurable  increase  of  the  moon's  angular  diameter  at  high 
altitudes  has  nothing  to  do  with  the  purely  subjective  illusion  which 
makes  the  disc  look  larger  to  us  when  near  the  horizon.  That  it  is  a 
mere  illusion  may  be  made  evident  bj*  simply  looking  through  a  dark 
glass  just  dense  enough  to  hide  the  horizon  and  intervening  land- 
scape. The  moon  or  sun  then  seems  to  shrink  at  once  to  normal 
dimensions. 

89.  Refraction. — Rays  of  light  have  their  direction  changed  by 
refraction  in  passing  through  the  air,  and  as  .the  direction  in  which  we 
see  a  body  is  that  in  which  its  light  .reaches  the  eye,  it  follows  that  this 
refraction  apparently  dis- 
places the  stars  and  all 
bodies  seen  through  the 
atmosphere.  So  far  as 
the  action  is  regular,  the 
effect  is  to  bend  the  rays 
directly  downwards,  and 
thus  to  make  the  objects 
appear  higher  in  the  sky. 
Refraction  increases  the 
altitude  of  a  celestial  ob- 
ject without  altering  the 
azimuth.  Like  parallax, 
it  is  zero  at  the  zenith 
and  a  maximum  at  the 
horizon  ;  but  it  follows  a 

different  law.*.  It  is^entirely  independent  of  the  distance  of  the 
object,  and  its  amount  varies  (nearly)  as  the  tangent  of  the  zenith 
distance  —  not  as  the  sine,  as  in  the  case  of  parallax. 


FIG.  35.  —  Atmospheric  Refraction. 


90.      This  approximate  law  of  the  refractioji  is  easily  proved. 

Suppose  in  Fig.  35  that  the  observer  at  0  sees  a  star  in  the  direction  OS, 
at  the  zenith  distance  ZOS  or  £.  The  light  has  reached  him  from  S>  by  a 
path  which  was  straight  until  the  ray  met  the  upper  surface  of  the  air  at  A. 
but  afterwards  curved  continually  downwards  as  it  passed  from  rarer  to 
denser  regions. 

We  know  that  the  atmosphere  is  very  shallow  as  compared  with  the  size 
of  the  earth,  and  it  is  exceedingly  rare  in  the  upper  portions,  so  that,  as 
far  as  concerns  refraction,  we  may  assume  that  the  point  A,  where  the  first 
perceptible  bending  of  the  ray  occurs,  is  not  more  than  fifty  miles  high, 
and  that  the  vertical  AZ'  is  sensibly  parallel  to  OZ ;  consequently,  also, 


64  CORRECTIONS  TO  OBSERVATIONS. 

that  all  the  successive  "  strata  of  equal  density  "  are  parallel  to  each  other  and 
to  the  upper  surface  of  the  air. 

[This  amounts  to  neglecting  the  earth's  curvature  between  0  and  JB.J 

The  true  zenith  distance  (as  it  would  be  if  there  were  no  refraction)  is 
ZDS',  which  equals  Z'A  Sr ;  and  since  the  refraction,  r,  may  be  denned  as 
the  difference  between  the  true  and  apparent  zenith  distances,  this  true 
zenith  distance  will  =  £  +  r. 

Now  from  optical  principles,  when  a  ray  of  light  passes  through  a 
medium  composed  of  parallel  strata,  the  final  direction  of  the  ray  is  the 
same  as  if  the  medium  had  throughout  the  density  of  the  last  stratum, 
and  therefore  the  final  direction,  SO,  will  be  the  same  as  if  all  the  air,  from 
A  down,  had  the  same  density  as  at  0,  with  the  same  index  of  refraction, 
n.  We  may  therefore  apply  the  law  of  refraction  directly  at  A,  and  write 
sin  Z'ASf=^  n  sin  BA C  (  =  ZOS),  or  sin  (£  +  r)  =  n  sin  £ ;  AC  being  drawn 
parallel  to  OS. 

Developing  the  first  member,  we  have 

sin  £  cos  r  +  cos  £  sin  r  =  n  sin  £. 

But  r  is  always  a  small  angle,  never  exceeding  40' ;  we  may  therefore  take 
cos  r=l.  Doing  this  and  transposing  the  first  term,  we  get 

cos  £  sin  r  —  n  sin  £  —  sin  £  =  (n  —  1)  sin  £. 
Whence,     sin  r  =  (n  —  1)  tan  £ ; 
or,  r"  =  (n  - 1)  206265  tan  £  (nearly). 


The  index  of  refraction  for  air,  at  zero  centigrade  and  a  barometric 
pressure  of  760™",  is  1.000294  ;  whence, 

r"  =  .000294  x  206265  x  tan£  =  60".6  tan  £. 

This  equation  holds  very  nearly  indeed  down  to  a  zenith  distance 
of  70°,  but  fails  as  we  approach  the  horizon.  For  rays  coming  nearly 
horizontal,  the  points  A  and  B  are  so  far  from  0  that  the  normal 
AZ'  is  no  longer  practically  parallel  to  OZ ;  and  many  of  the  other 
fundamental  assumptions  on  which  the  formula  is  based  also  break 
down. 

At  the  horizon,  where  £  =  90°  and  tan  £  =  infinity,  the  formula 
would  give  sin  r  =  infinity  also ;  an  absurdity,  since  no  sine  can 
exceed  unity.  The  refraction  there  is  really  about  37',  under  the 
circumstances  of  temperature  and  pressure  above  indicated. 


REFRACTION.  66 

91 .  Effect  of  Temperature  and  Barometric  Pressure.  — The  index 
of  refraction  of  air  depends  of  course  upon  its  temperature  and  pressure. 
As  the  air  grows  warmer,  its  refractive  power  decreases;  as  it  grows 
denser,  the  refraction  increases.     Hence,  in  all  precise  observations  of 
the  altitude   (or  zenith   distance),  it  is   necessary   to  note  both  the 
thermometer  and  the  barometer,  in  order  to   compute   the  refraction 
with  accuracy.     For  rough  work,  like  ordinary  sextant  observations, 
it  will  answer  to   use  the  "  mean  refraction,"   corresponding  to  an 
average  state  of  things.     See  Appendix,  Table  VIII. 

Tables  of  Refraction.  —  The  exact  computation  of  the  refraction  is  best 
effected  by  special  tables  for  the  purpose  ;  of  these,  Bessel's  tables  are 
the  most  convenient,  best  known,  and  probably  even  yet  the  most  accurate. 
It  must  be  always  borne  in  mind,  however,  that  from  the  action  of  wind  and 
other  causes  the  condition  of  the  air  along  the  path  of  the  ray  is  seldom  per- 
fectly normal ;  in  consequence,  the  actual  refraction  in  any  given  case  is  lia- 
ble to  differ  from  the  computed  by  as  much  as  one  or  even  two  per  cent. 
No  amount  of  care  in  observation  can  evade  this  difficulty ;  the  only  remedy 
is  a  sufficient  repetition  of  observations  under  varying  atmospheric  condi- 
tions. Observations  at  an  altitude  below  10°  or  15°  are  never  much  to  be 
trusted. 

Lateral  Refraction.  —  When  the  air  is  much  disturbed,  sometimes  ob- 
jects are  displaced  horizontally  as  well  as  vertically.  Indeed,  as  a  general 
rule,  when  one  looks  at  a  star  with  a  large  telescope  and  high  power,  it  will 
seem  to  "  dance  "  more  or  less  —  the  effect  of  the  varying  refraction  which 
continually  displaces  the  image. 

92.  Effect  on  the  Time  of  Sunrise  and  Sunset.  — The  horizontal  re- 
fraction, ranging  as  it  does  from  32'  to  40',  according  to  temperature, 
is  always  somewhat  greater  than  the  diameter  of  either  the  sun  or 
the  moon.     At  the  moment,  therefore,  when  the  sun's  lower  limb 
appears  to  be  just  rising,  the  whole  disc  is  really  below  the   plane 
of   the   horizon ;    and   the   time   of   sunrise  in  our  latitudes  is  thus 
accelerated  from  two  to  four  minutes,  according  to  the  inclination  of 
the  sun's  diurnal  circle  to  the  horizon,  which  inclination  varies  with 
the  time  of  the  year.     Of   course,  sunset   is   delayed   by  the  same 
amount,  and   thus   the   day  is   lengthened   by  refraction   from  four 
to  eight  minutes,  at  the  expense  of  the  night. 

93.  Effect  on  the  Form  and  Size  of  the  Discs  of  the  Sun  and  Moon. 

—  Near  the  horizon  the  refraction  changes  very  rapidly.     While  un- 
der ordinary  summer  temperature  it  is  about  35'  at  the  horizon,  it  is 


66  DETERMINATION   OF  THE  REFRACTION. 

only  29'  at  an  elevation  of  half  a  degree  ;  so  that,  as  the  sun  or  moon 
rises,  the  bottom  of  the  disc  is  lifted  6f  more  than  the  top,  and  the 
vertical  diameter  is  thus  made  apparently  about  one-fifth  part  shorter 
than  the  horizontal.  This  distorts  the  disc  into  the  form  of  an  oval, 
flattened  on  the  under  side.  In  cold  weather  the  effect  is  much  more 
marked.  As  the  horizontal  diameter  is  not  at  all  increased  by  the 
refraction,  the  apparent  area  of  the  disc  is  notably  diminished  by  it ; 
so  that  it  is  evident  that  refraction  cannot  be  held  in  any  way  re- 
sponsible for  the  apparent  enlargement  of  the  rising  luminary. 

94.  Determination  of  the  Refraction.  —  1.  Physical  Method. 
Theory  furnishes  the  law  of  astronomical  refraction,  though  the 
mathematical  expression  becomes  rather  complicated  when  we  attempt 
to  make  it  exact.  In  order,  therefore,  to  determine  the  astronomical 
refraction  under  all  possible  circumstances,  it  is  only  necessary  to 
determine  the  index  of  refraction  of  air, and  its  variations  with  tem- 
perature and  pressure, by  laboratory  experiments,  and  to  introduce  the 
constants  thus  obtained  into  the  formulae.  It  is  difficult,  however,  to 
make  these  determinations  with  the  necessary  precision.  In  fact,  at 
present  our  knowledge  of  the  constants  of  air  rests  mainly  on  astro- 
nomical work. 

2.  By  Observations  of  Circumpolar  Stars.  At  an  observatory  whose 
latitude  exceeds  45°  select  some  star  which  passes  through  the  zenith 
at  the  upper  culmination.  (Its  declination  must  equal  the  latitude  of 
the  observatory.)  It  will  not  be  affected  by  refraction  at  the  zenith, 
while  at  the  lower  culmination,  twelve  hours  later,  it  will.  With  the 
meridian  circle  observe  its  polar  distance  in  both  positions,  determin- 
ing the  "  polar  point "  of  the  circle  as  described  on  pp.  46-47.  If  the 
polar  point  were  not  itself  affected  by  refraction,  the  simple  differ- 
ence between  the  two  results  for  the  star's  polar  distance,  obtained 
from  the  upper  and  lower  observations,  would  be  the  refraction  at 
the  lower  point. 

As  a  first  approximation,  however,  we  may  neglect  the  refraction 
at  the  pole,  and  thus  obtain  a  first  approximate  lower  refraction. 
By  means  of  this  we  may  compute  an  approximate  polar  refraction, 
and  so  get  a  first  "corrected  polar  point."  With  this  compute  a 
second  approximate  lower  refraction,  which  will  be  much  more  nearly 
right  than  the  first ;  this  will  give  a  second  "  corrected  polar  point"  ; 
this  will  in  turn  give  us  a  third  approximation  to  the  refraction  ;  and 
so  on.  But  it  would  never  be  necessary  to  go  beyond  the  third,  as 
the  approximation  is  very  rapid.  If  the  star  does  not  go  exactly 


TWILIGHT.  67 

through  the  zenith,  it  is  only  necessary  to  compute  each  time  approxi- 
mate refractions  for  its  upper  observation,  as  well  as  for  the  polar 
point. 

At  present,  however,  the  refraction  is  so  well  known  that  the 
method  actually  used  is  to  form  "equations  of  condition"  from  the 
observations  of  the  altitude  of  known  stars  under  varying  circum- 
stances, and  from  these  to  deduce  such  corrections  to  the  star  places 
and  refraction  constants  as  will  best  harmonize  the  whole  mass  of 
material. 

95.  3.  By  Observations  of  the  Altitudes  of  Equatorial  Stars  made  at  an  Ob- 
servatory near  the  Equator.  For  an  observer  on  the  equator,  stars  on  the 
celestial  equator  (8  —  0)  will  come  to  the  meridian  at  the  zenith,  and  will  rise 
and  fall  vertically,  with  a  motion  strictly  proportional  to  the  time  ;  the  true  zenith 
distance  of  the  star  at  any  moment  being  just  equal  to  its  hour-angle  in 
degrees.  We  have  only,  then,  to  observe  the  apparent  zenith  distance  of  a 
star  with  the  corresponding  time,  and  the  refraction  comes  out  directly. 

If  the  station  is  not  exactly  on  the  equator,  and  if  the  star's  declination  is 
not  exactly  zero,  it  is  only  necessary  to  know  the  latitude  and  declination 
approximately  in  order  to  get  the  refraction  very  accurately :  a  considerable 
error  in  either  latitude  or  declination  will  affect  the  result  but  slightly. 

4.  The  French  astronomer  Loewy  devised  a  new  method  which  has 
proved  excellent.  He  puts  a  pair  of  reflectors,  inclined  to  each  other  at  a 
convenient  angle  of  from  45°  to  50°  (a  glass  wedge  with  silvered  sides),  in  front 
of  the  object-glass  of  an  equatorial.  This  will  bring  to  the  eye  two  rays 
which  make  a  strictly  constant  angle  with  each  other,  and  there  is  no  diffi- 
culty in  finding  pairs  of  stars  so  situated  that  their  images  will  come  into 
the  field  of  view  together.  Now,  were  it  not  for  refraction,  thes6  images 
would  always  keep  their  relative  position  unchanged,  notwithstanding  the 
diurnal  motion ;  but  on  account  of  the  changes  in  the  refraction,  as  one  star 
rises  and  the  other  falls,  they  will  shift  in  the  field,  and  micrometric  meas- 
ures will  determine  the  shifting,  and  so  the  refraction,  with  great  precision. 

96.  Twilight.  —  (Although  this  subject  is  outside  the  main  purpose  of  this 
chapter,  which  deals  with  corrections  to  be  applied  to  astronomical  observations,  we 
treat  it  here  because,  like  refraction,  it  is  a  purely  atmospheric  phenomenon,  and 
finds  no  other  more  convenient  place.) 

Twilight,  the  illumination  of  the  sky  which  begins  before  sunrise, 
and  continues  after  sunset,  is  caused  by  the  reflection  of  light  to  the 
observer  from  the  upper  regions  of  the  earth's  atmosphere.  It  is  not 
yet  certain  whether  this  is  due  to  reflection  from  foreign  matter  in  the 
air,  such  as  minute  crystals  of  ice  and  salt,  particles  of  dust  of 
various  kinds,  and  infinitesimal  drops  of  water,  or  whether  the  pure 
gases  themselves  have  some  power  of  reflecting  light.  There  is  no 


68 


TWILIGHT. 


doubt,  however,  that  air,  under  the  ordinary  conditions,  possesses 
considerable  power  of  reflection  ;  so  that,  as  long  as  any  air  upon 
which  the  sun  is  shining  is  visible  to  the  observer,  it  will  send  him 
more  or  less  light,  and  appear  illuminated. 

Suppose  the  atmosphere  to  have  the  depth  indicated  in  the  figure. 
Then,  if  the  sun  is  at  S,  Fig.  36,  it  will  just  have  set  to  an  observer 
at  1,  but  all  the  air  within  his  range  of  vision  will  still  be  illuminated. 
When,  by  the  earth's  rotation,  he  has  been  transported  to  2,  he  will 
see  the  "  twilight  bow  "  rising  in  the  east,  a  faintly  reddish  arc 
separating  the  illuminated  part  of  the  sky  from  the  darkened  part 
below,  which  lies  in  the  shadow  of  the  earth.  When  he  reaches  3, 
the  western  half  of  the  sky  alone  remains  bright,  but  the  arc  of 


FIG.  36.— Twilight. 


separation  between  the  light  and  darkness  has  become  vague  and 
indefinite;  when  he  reaches  4,  only  a  glow  remains  in  the  west; 
and  when  he  comes  to  5,  night  closes  in  on  him.  Nothing  remains 
in  sight  on  which  the  sun  is  shining. 

97.  Duration  of  Twilight.  —  This  depends  upon  the  height  of  the  atmos- 
phere, and  the  angle  at  which  the  sun's  diurnal  circle  cuts  the  horizon.  It  is 
found  as  a  matter  of  observation,  not  admitting,  however,  of  much  precision, 
that  twilight  lasts  until  the  sun  has  sunk  about  18°  below  the  horizon;  that 
is  to  say,  the  angle  1  C  5  in  the  figure  is  about  18°. 

The  time  required  to  reach  this  point  in  latitude  40°  varies  from  two 
hours  at  the  longest  days  in  summer,  to  one  hour  thirty  minutes  about  Oct. 
12  and  March  1,  when  it  is  least.  At  the  winter  solstice  it  is  about  one 
hour  and  thirty-five  minutes. 

In  higher  latitudes  the  twilight  lasts  longer,  and  the  variation  is  more 
considerable  :  the  date  of  the  minimum  also  shifts. 

Near  the  equator  the  duration  is  shorter,  hardly  exceeding  an  hour  at  the 


HEIGHT   OF   THE   ATMOSPHERE.  69 

sea-level;  while  at  high  elevations  (where  the  amount  of  air  above  the 
observer's  level  is  less)  it  becomes  very  brief.  At  Quito  and  Lima  it  is 
said  not  to  last  more  than  twenty  minutes.  Probably,  also,  in  mountain 
regions  the  clearness  of  the  air,  and  its  purity  contribute  to  the  effect. 

98.  Height  of  the  Atmosphere. — It  is  evident  from  the  figure  that 
at  the  moment  twilight  ceases,  the  last  visible  portion  of  illuminated 
air  is  at  the  top  of  the  atmosphere,  and  just  half-way  between  the  ob- 
server and  the  nearest  point  where  the  sun  is  setting.    If  the  whole  arc 
1,  5  is  18°,  1,  3  is  9°  :  then  calling  the  height  of  the  atmosphere  H  and 
the  earth's  radius  J?,  and  neglecting  refraction  (i.e.,  supposing  the  lines 
1  m  and  5  m  to  be  straight) ,  we  have  from  the  right-angled  triangle 
1  Cm,  Cm  =  1  G  X  sec 9°,  or  R  +  H=  R  x  sec  9°  ;  whence  H=  R 
(sec  9°—  1)  =  0.0125  R,  or  almost  exactly  fifty  miles.     This  must 
be  diminished  about  one-fifth  part  on  account  of   the   curvature   of 
the  lines  1m  and  5m  by  refraction,  making  the  height  of  the  atmos- 
phere about  fort}'  miles. 

The  result  must  not,  however,  be  accepted  too  confidently.  It  only 
proves  that  we  get  no  sensible  twilight  illumination  from  air  at  a 
greater  height :  above  that  elevation  the  air  is  either  too  rare,  or  too 
pure  from  foreign  particles,  to  send  us  any  perceptible  reflection. 
There  is  abundant  evidence  from  the  phenomena  of  meteors  that  the 
atmosphere  extends  to  a  height  of  100  miles  at  least,  and  it  cannot 
be  asserted  positively  that  it  has  any  definite  upper  limit. 

99.  Aberration.  —  There  is  yet  one  more  correction  which  has  to  be 
applied  in  order  to  get  the  true  direction  of  the  line  which  at  the  instant  of 
observation  joins  the  eye  of  the  observer  to  the  star  he  is  pointing  at.     The 
aberration  of  light  is  an  apparent  displacement  of  the  object  observed,  due 
to  the  combination  of  the  earth's  orbital  motion  with  the  progressive  motion 
of  light.     It  can  be  better  discussed,  however,  in  a  different  connection  (see 
Chap.  VI.),  and  we  content  ourselves  with  merely  mentioning  it  here. 


EXERCISES  ON  CHAPTER  I. 

1.  What  point  in  the  celestial  sphere  has  both  its  right  ascension  and 
declination  zero  ? 

2.  What  are  the  celestial  latitude  and  longitude  of  this  point? 

3.  What  are  the  hour-angle  and  azimuth  of  the  zenith  ? 

4.  What  angle  does  the  celestial  equator  make  with  the  horizon  at  a 
place  in  latitude  40°? 


70  QUESTIONS. 

5.  Name  the  fourteen  principal  points  on  the  celestial  sphere,  zenith, 
poles,  equinoxes,  etc. 

6.  What  important  circles  on  the  celestial  sphere  have  no  correlatives  on 
the  surface  of  the  earth? 

7.  What  are  the  approximate  right  ascension  and  declination  of  the  sun 
on  September  22  ? 

8.  If  a  certain  star  culminates  (comes  to  the  meridian)  at  eight  o'clock 
to-night,  at  what  time  will  it  culminate  ten  days  hence  ? 

9.  What  is  the  altitude  of  the  sun  on  March  21  at  noon  for  an  observer 
in  latitude  40°  30'? 

10.  On  March  21,  one  hour  after  sunset,  whereabouts  in  the  sky  would 
be  the  position  of  a  star  having  a  right  ascension  of  7  hours  and  a  decli- 
nation of  +  40°,  the  observer  being  in  latitude  45°  ? 


EXERCISES  ON  CHAPTER  II. 

1.  If  a  firefly  were  to  light  on  the  object-glass  of  a  telescope,  what  would 
be  the  appearance  to  an  observer  looking  through  the  instrument  ? 

2.  If  a  triangular  piece  of  paper  were  pasted  on  the  object-glass  of  a 
telescope  pointed  at  the  moon,  how  would  it  affect  the  appearance  of  the 
moon  as  seen  by  the  observer  ? 

3.  If  a  certain  eye-piece  gives  a  magnifying  power  of  60  when  used  with 
a  telescope  of  5  feet  focal  length,  what  power  will  it  give  when  used  on  a 
telescope  of  30  feet  focus  ? 

4.  If  the  wires  of  a  micrometer  (Fig.  29,  Art.  73)  are  so  set  that,  used 
with  a  telescope  of  10  feet  focal  length,  a  star  moving  along  the  right 
ascension  wire  will  occupy  15  seconds  in  passing  from  d  to  e,  how  long  will 
it  take  when  the  micrometer  is  transferred  to  a  telescope  of  50  feet  focus  ? 

5.  What  is  theoretically  the  angular  distance  between  the  centres  of  two 
star  discs  which  are  just  barely  separated  by  the  Yerkes  telescope  of  40 
inches  aperture? 

.   6.   What  is  magnitude  of  1"  on  a  graduated  circle  of  2  feet  diameter? 

7.    Why  is  it  important  that  the  two  pivots  of  a  transit  instrument 
should  be  of  exactly  the  same  diameter? 


QUESTIONS.  71 


EXERCISES  ON  CHAPTER  III. 

1.  What  is  the  approximate  dip  of  the  horizon  from  a  hill  900  feet  high? 

2.  How  high  must  a  mountain  be  in  order  that  the  dip  of  the  horizon 
from  its  summit  may  be  2°? 

3.  Does  atmospheric  refraction  increase  or  decrease  the  apparent  size  of 
the  sun's  disc  when  it  is  near  the  horizon  ?     Why  ? 

4.  Assuming  the  horizontal  parallax  of  the  sun  at  8.8",  what  is  the 
horizontal  parallax  of  Mars  when  nearest  us  at  a  distance  of  0.378  astro- 
nomical units  ? 

5.  What  is  the  greatest  apparent  diameter  of  the  earth  as  seen  from 
Mars? 

6.  What  is  the  horizontal  parallax  of  Jupiter  when  at  a  distance  of  6 
astronomical  units  ? 

7.  What  is  the  lowest  latitude  where  twilight  can  last  all  night?     Can 
it  do  so  at  New  York  ?     At  London  ?     At  Edinburgh  ? 


72  PROBLEMS    OF   PRACTICAL   ASTRONOMY. 


CHAPTER   IV. 

PROBLEMS  OF  PRACTICAL  ASTRONOMY,  LATITUDE,  TIME, 
LONGITUDE,  AZIMUTH,  AND  THE  RIGHT  ASCENSION  AND 
DECLINATION  OF  A  HEAVENLY  BODY. 

100.  THERE  are  certain  problems  of  Practical  Astronomy  which 
have  to  be  solved  in  obtaining  the  fundamental  facts  from  which 
we  deduce  our  knowledge  of  the  earth's  form  and  dimensions,  and 
other  astronomical  data. 

The  first  of  these  problems  is  that  of  the  LATITUDE. 

DEFINITION  OF  LATITUDE. 

The  latitude  (astronomical)  of  a  place  (Art.  30)  is  simply  the  alti- 
tude of  the  celestial  pole  (Polhohe),  or,  what  comes  to  the  same  thing, 
as  is  evident  from  Fig.  7  (Art.  33),  it  is  the  declination  of  the  zenith. 
It  may  also  be  defined,  from  the  mechanical  point  of  view,  as  the 
angle  between  the  plane  of  the  earth's  equator  and  the  observer's  plumb- 
line  or  vertical. 

Neither  of  these  definitions  assumes  anything  as  to  the  form  of  the  earth. 
This  astronomical  latitude  is  seldom  identical  with  the  geocentric,  or  even 
with  the  geodetic  or  charto graphical  latitude  of  a  place  —  the  latitude  used  in 
accurate  mapping.  It  is,  however,  the  only  kind  of  latitude  which  can  be 
directly  determined  from  astronomical  observations,  and  its  determination 
is  one  of  the  most  important  operations  of  Economic  Astronomy. 

101.  Determination  of  Latitude.  —  First :  By  Circumpolars.    The 
•most  obvious  method  of  determining  the  latitude  is  to  observe,  with 
the  meridian  circle  or  some  analogous  instrument,  the  altitude  of  a 
circumpolar  star  at  its  upper  culmination,  and  again,  twelve  hours 
later,  at  its  lower.     Each  of  the  observations  must  be  corrected  for 
refraction,  and  then  the  mean  of  the  two  corrected  altitudes  will  be  the 
latitude. 

This  method  has  the  advantage  of  being  an  independent  one  ;  i.e.,  it  does 
not  require  any  data  (such  as  the  declination  of  the  stars  used)  to  be  accepted 
on  the  authority  of  previous  observers.  But  to  obtain  much  accuracy  it 
requires  considerable  time  and  a  large  fixed  instrument.  In  low  latitudes 
the  refraction  is  also  very  troublesome. 


LATITUDE.  73 

102.  Second  :  By  the  meridian  altitude  or  zenith  distance  of  a 
body  of  known  declination. 

In  Fig.  37  the  semicircle  AQPB  is  the  meridian,  Q  and  P  being 
respectively  the  equator  and  the  pole,  and  Z  the  zenith.  QZ  is 
the  declination  of  the  zenith,  or  the  observer's  latitude  (=  PB=  <£). 
Suppose  now  that  we  observers  (=  £s),  the  zenith  distance  of  a 
star  s  (south  of  the  zenith),  as  it  crosses  the  meridian,  and  that  its 
declination  Qs  (=  88)  is  known  ;  then  evidently  <f>  =  8,  +  £«• 

In  the  same  way,  if  the  star  were  at    n,  between  zenith  and  pole, 


If  we  use  the  meridian  circle,  we  can  always  select  stars  that  pass  near 
the  zenith  where  the  refraction  will  be  small  ;  moreover,  we  can  select  them 
in  such  a  way  that  some  will  be  as  much  north  of  the  zenith  as  others  are 
south,  and  thus  eliminate  the  refraction  errors.  But  we  have  to  get  our  star 
declinations  out  of  catalogues  made  by  previous  observers,  and  so  the  method 
is  not  an  independent  one. 

103.  At  Sea  the  latitude  is  usually"  obtained  by  observing  with  the  sex- 
tant the  sun's  maximum  altitude,  which 

of  course  occurs  at  noon.  Since  at  sea 
it  is  seldom  that  one  knows  beforehand 
precisely  the  moment  of  local  noon, 
the  observer  takes  care  to  begin  to  ob- 
serve the  sun's  altitude  some  ten  or 
fifteen  minutes  earlier,  repeating  his 

FIG.  37.  —  Determination  of  Latitude. 

observations  every  minute  or  two.     At 

first  the  altitude  will  keep  increasing,  but  immediately  after  noon  it 
will  begin  to  decrease.  The  observer  uses  therefore  the  maximum1 
altitude  obtained,  which,  corrected  for  refraction,  parallax,  semi- 
diameter,  and  dip  of  the  horizon,  will  give  him  the  true  latitude  of 
his  ship,  b}^  the  formula  <£  =  8  ±  £. 

104.  Third:  By  Circum-meridian  Altitudes.  —  If  the  observer  knows       *• 
his  time  with  reasonable  accuracy,  he  can  obtain  his  latitude  from  observa- 
tions made  when  the  body  is  near  the  meridian,  with  practically  the  same 
precision  as  at  the  moment  of  meridian  passage.     It  would  take  us  a  little 

1  On  account  of  the  sun's  motion  in  declination,  and  the  northward  or  south- 
ward motion  of  the  ship  itself,  the  sun's  maximum  altitude  is  usually  attained 
not  precisely  on  the  meridian,  but  a  few  seconds  earlier  or  later.  This  requires  a 
slight  correction  to  the  deduced  latitude,  explained  in  books  on  Navigation  or 
Practical  Astronomy. 


74 


PROBLEMS   OF   PRACTICAL   ASTRONOMY, 


too  far  to  explain  the  method  of  reduction,  which  is  given  with  the  necessary 
tables  in  all  works  on  Practical  Astronomy.  The  great  advantage  of  this 
method  is  that  the  observer  is  not  restricted  to  a  single  observation  at  each 
meridian-passage  of  the  sun  or  of  the  selected  star,  but  can  utilize  the  half- 
hours  preceding  and  following  that  moment.  The  meridian-circle  cannot 
be  used,  as  the  instrument  must  be  such  as  to  make  extra-meridian  observa- 
tions possible.  Usually  the  sextant  or  universal  instrument  is  employed. 
This  method  is  much  used  in  the  French  and  German  geodetic  surveys. 


105.  Fourth  :  The  Zenith  Telescope  Method.  —  (Sometimes  known  as 
the  American  method,  because  first  practically  introduced  by  Captain  Talcott 
of  the  United  States  Engineers,  in  a  boundary  survey  in  1845.) 

The  essential  characteristic  of  the  method  is  the  micrometric 
measurement  of  the  difference  between  the  nearly  equal  zenith  dis- 

tances  of  two  stars  which  culminate 
within  a  few  minutes  of  each  other, 
one  north  and  the  other  south  of  the 
zenith,  and  not  very  far  from  it  :  such 
pairs  of  stars  can  always  be  found. 
When  the  method  was  first  introduced, 
a  special  instrument,  known  as  the 
zenith  telescope,  was  generally  em- 
ployed, but  at  present  a  simple  transit 
instrument,  with  declination  microm- 
eter, and  a  delicate  level  attached  to 
the  telescope  tube,  is  ordinarily  used* 
The  telescope  is  set  at  the  proper 
altitude  for  the  star  which  first  comes 
to  the  meridian,  and  the  "latitude 
level,"  as  it  is  called,  is  set  horizontal  j 
as  the  star  passes  through  the  field  of 
view  its  distance  north  or  south  of  the 
central  wire  is  measured  by  the  micrometer.  The  instrument  is  then 
reversed,  and  so  set  by  turning  the  telescope  up  or  down  (without, 
however,  disturbing  the  angle  9  (Fig.  38)  between  the  level  and  tele- 
scope), that  the  level  is  again  horizontal.  After  this  reversal  and 
adjustment,  the  telescope  tube  is  then  evidently  elevated  at  exactly 
the  same  angle,  £,  as  before,  but  on  the  opposite  side  of  the  zenith. 
As  the  second  star  passes  through  the  field,  we  measure  with  the 
micrometer  its  distance  north  or  south  of  the  centre  of  the  field  ; 
the  comparison  of  the  two  micrometer  measures  gives  the  difference 
of  the  two  zenith  distances. 


FIG.  38.  —  Principle  of  the  Zenith 
Telescope. 


LATITUDE. 


75 


From  Fig.  37  we  have 


for  star  south  of  zenith,  <f>  =  8S  +  £8 ; 
for  star  north  of  zenith,  <f>  =  8n  —  £n. 

Adding  the  two  equations  and  dividing  by  2,  we  have 


The  declinations,  Ss  and  8n,  are  given  in  the  star  catalogue,  and  the 
micrometer  gives  (8S  —  8n).  Usually,  also,  small  corrections,  seldom  reach- 
ing 1",  must  be  addended  for  differential  refractions,  and  for  level-change, 
if  any. 

The  great  advantage  of  the  method  consists  in  its  dispensing  with  a 
graduated  circle,  and  in  avoiding  almost  wholly  the  errors  due  to  refraction. 
Forty  years  ago  it  was  not  always  easy  to  find  accurate  determinations  of 
the  declinations  of  the  stars  employed,  but  at  present  this  difficulty  has 
practically  disappeared,  so  that  this  method  of  determining  the  latitude  is 
now  not  only  the  most  convenient  and  rapid, 
but  is  quite  as  precise  as  any,  if  the  level  is 
sufficiently  sensitive.  Evidently  the  accuracy 
depends  upon  the  exactness  with  which  the  « 
level  measures  the  slight,  but  inevitable,  dif- 
ference between  the  inclinations  of  the  instru- 
ment when  pointed  on  the  two  stars.  In  Dr. 
Chandler's  "  Almucantar,"  an  instrument  used 
for  the  same  purpose,  the  telescope  preserves 
its  constant  inclination  automatically,  by  being 
mounted  upon  a  base  which  floats  in  mercury, 
thus  dispensing  with  the  level. 


FIG.  39. 
Latitude  by  Prime  Vertical  Transits. 


106.  Fifth  :  By  the  Prime  Vertical  Instrument  (p.  42). — We  observe 
simply  the  moment  when  a  known  star  passes  the  prime  vertical  on  the 
eastern  side,  and  again  upon  the  western  side.  Half  the  interval  will  give 
the  hour-angle  of  the  star  when  on  the  prime  vertical ;  i.e.,  the  angle  ZPS 
in  Fig.  39,  where  Z  is  the  zenith,  P  the  pole,  and  SZS'  the  prime  vertical. 
The  distance  PS  of  the  star  from  the  pole  is  the  complement  of  the  star's 
declination  ;  and  PZ  is  the  complement  of  the  observer's  latitude.  Since 
the  prime  vertical  is  perpendicular  to  the  meridian  at  the  zenith,  the  tri- 
angle PZS  will  be  right-angled  at  Z,  and  from  Napier's  rule  of  circular 
parts  (taking  ZPS  as  the  middle  part)  we  shall  have 


or 
whence 


cos  ZPS  =  tan  PZ  cot  PS, 
cos  t          =  cot  (j>  tan  S ; 
tan  =  tan  S  sec  t. 


76 


PROBLEMS    OF   PRACTICAL   ASTRONOMY. 


If  8  nearly  equals  <£,  t  will  be  small,  and  a  considerable  error  in  the  obser- 
vation of  t  will  then  produce  very  little  change  in  its  secant  or  in  the  com- 
puted latitude. 

The  observations  are  not  so  convenient  and  easy  as  in  the  case  of  the 
zenith  telescope,  and  the  number  of  stars  available  is  less;  but  the  method 
presents  the  great  advantage  of  requiring  nothing  but  an  ordinary  transit 
instrument,  without  any  special  outfit  of  micrometer  and  latitude  level. 
It  also  entirely  evades  the  difficulties  caused  by  refraction. 

107.  Sixth:  By  the  Gnomon. — The  ancients  had  no  instruments 
such  as  we  have  hitherto  described,  and  of  course  could  not  use  any 
of  the  preceding  methods  of  finding  the  latitude.  They  were,  how- 
ever, able  to  make  a  very  respectable  approximation  by  means  of  the 
simplest  of  all  astronomical  instruments,  the  gnomon.  This  is  merely 
a  vertical  shaft  or  column  of  known  height  erected  on  a  perfectly 

horizontal  plane ;  and  the 
observation  consists  in  not- 
ing the  length  of  the  shadow 
cast  at  noon  at  certain  times 
of  the  year. 

Suppose,  for  instance,  that 
on  the  day  of  the  summer 
solstice,  at  noon,  the  length 
of  the  shadow  is  AC,  Fig.  40. 
The  height  AB  being  given, 
we  can  easily  compute  in 
the  right-angled  triangle  the 
angle  ABC,  which  equals 
SBZ,  the  sun's  zenith  dis- 
tance when  farthest  north.  Again  observe  the  length  AD  of  the 
shadow  at  noon  of  the  shortest  day  in  winter,  and  compute  the  angle 
ABD,  which  is  the  sun's  corresponding  zenith  distance  when  farthest 
south.  Now,  since  the  sun  travels  equal  distances  north  and  south 
of  the  celestial  equator,  the  mean  of  the  two  results  will  give  the 
angular  distance  between  the  equator  and  the  zenith ;  i.e.,  the  decli- 
nation of  the  zenith,  which  (Art.  100)  is  the  latitude  of  the  place. 

The  method  is  an  independent  one,  like  that  by  the  observation  of  cir- 
cumpolar  stars,  requiring  no  data  except  those  which  the  observer  determines 
for  himself.  Evidently,  however,  it  does  not  admit  of  much  accuracy,  since 
the  penumbra  at  the  end  of  the  shadow  makes  it  impossible  to  measure  its 
length  very  precisely. 

It  should  be  noted  that  the  ancients,  instead  of  designating  the  position 


A  C        E 

FIG.  40.  —  Latitude  by  the  Gnomon. 


DETERMINATION    OF    TIME.  77 

of  a  place  by  means  of  its  latitude,  used  its  climate  instead  ;  the  climate 
(from  /cXfyia)  being  the  slope  of  the  plane  of  the  celestial  equator,  the  angle 
AEB,  which  is  the  complement  of  the  latitude. 

For  the  use  of  the  gnomon  in  determining  the  obliquity  of  the  ecliptic 
and  the  length  of  the  year,  see  Art.  176.  Many  of  the  Egyptian  obelisks 
are  known  to  have  been  used  for  astronomical  observations  and  were  prob- 
ably erected  mainly  for  that  purpose. 

For  numerous  other  methods  of  determining  the  Latitude,  see 
Chauvenet's  Practical  Astronomy. 

108,  Variation  of  Latitude  and  Motion  of  the  Poles.  —  It  has  long 
been  doubted  whether  latitudes  are  strictly  constant.  They  cannot 
be  so  if  the  axis  of  the  earth  shifts  its  position  within  the  globe, 
for  then  the  poles  must  also  move,  and  the  latitudes  of  places  will 
change  correspondingly.  Some  have  supposed  that  in  the  past 
there  have  been  great  changes  of  this  kind,  and  have  sought  thus  to 
explain  certain  geological  epochs,  as  for  instance  the  glacial  and  the 
carboniferous.  But  thus  far  no  confirmatory  evidence  of  such  dis- 
placement has  appeared  ;  nor  is  there  yet  any  absolute  evidence 
of  certain  slow,  continuous,  "secular"  changes  which  have  been 
strongly  suspected. 

Theoretically  any  alteration  in  the  arrangement  of  the  matter 
of  the  earth,  by  elevation,  subsidence,  transportation,  or  denudation, 
must  almost  necessarily  disturb  the  axis  to  some  extent.  The  ques- 
tion is  merely  whether  we  can  observe  with  sufficient  accuracy  to 
detect  the  changes.  Within  the  past  few  years  this  limit  has  been 
reached,  and  we  now  have  conclusive  proof  of  slight,  but  unquestion- 
able, periodical  "variations  of  latitude"  The  first  satisfactory  evi- 
dence of  such  variations  was  obtained  from  observations  made  at 
Berlin  (by  Kiistner)  and  at  other  German  stations  in  1888  and  1889. 
The  result  has  since  been  abundantly  confirmed  by  observations  in 
Russia,  France,  England,  the  United  States,  and  in  the  Sandwich 
Islands.  Moreover,  Dr.  S.  C.  Chandler  of  Cambridge  (U.  S.),  by  a 
brilliant  and  laborious  series  of  investigations,  finds  the  same  varia- 
tions exhibited  clearly  in  almost  every  extended  body  of  reliable 
observations  made  since  1750.  From  the  whole  mass  of  evidence 
he  concludes  that  the  movement  of  the  pole  is  composed  of  two  mo- 
tions, one  an  annual  revolution  in  a  narrow  ellipse  about  30  feet 
long,  but  varying  in  form  and  position,  the  other,  a  revolution  in  a 
circle  about  26  feet  in  diameter,  with  a  period  of  about  428  days  : 
both  motions  are  counter-clock-wise.  The  resultant  motion  appears 
very  irregular,  and  varies  greatly  from  year  to  year. 


78  PROBLEMS   OF   PRACTICAL   ASTRONOMY. 

Fig..  40*  (at  the  end  of  the  chapter,  page  95)  represents  the  actual  motion 
from  1890  to  1895  as  deduced  by  Albrecht  from  all  available  observations. 

The  polar  displacements  produce  also  slight  changes  of  Azimuth  in  geo- 
detic lines,  as  has  been  actually  observed  at  Pulkowa  ;  and  a  minute  tide 
(only  a  fraction  of  an  inch),  which  theory  indicates  as  a  necessary  conse- 
quence of  shiftings  of  the  earth's  axis,  has  been  detected  in  the  Pacific  and 
Atlantic  oceans,  and  on  the  coast  of  Holland.  The  annual  component  of 
this  polar  motion  is  very  likely  due  to  meteorological  causes  which  follow 
the  seasons,  such  as  the  deposit  and  melting  of  snow  and  ice.  The  expla- 
nation of  the  428  day  component  is  not  yet  entirely  clear,  and  its  discussion 
would  take  us  too  far. 

It  is  likely  also  that  irregular  disturbances,  due  to  various  causes,  occa- 
sionally modify  the  regular  periodic  motions. 


TIME   AND    ITS   DETEKMINATION. 

109.  One  of  the   most   important   problems   presented  to  the 
astronomer  is  the  determination  of  Time.     By  universal  consent  the 
apparent  rotation  of  the  heavens  is  made  to  furnish  the  standard, 
and  the  determination  of  time  is  effected  by  ascertaining  by  obser- 
vation the  hour-angle  of  the  object  selected  to  mark  the  beginning  of 
the  day  by  its  transit  across  the  meridian.    *In  practice  three  kinds  of 
time  arjj.  now  recognized,  viz.,  sidereal  time,  apparent  solar  time,  and 
mean  solar  time. 

(For  definition  of  hour-angle,  see  Art.  24.) 

110.  Sidereal  Time.  —  As  has  already  been  explained  (Art.  26), 
the  sidereal  time  at  any  moment  is  the  hour-angle  of  the  vernal  equi- 
nox at  that  moment;  or,  what  comes  to  the  same  thing,  though  it 
sounds  differently,  it  is  the  time  marked  by  a  clock  which  is  so  set  and 
adjusted  as  to  show  noon,  or  Oh  00m  00s,  at  each  transit  of  the  vernal 
equinox.     The  sidereal  day,  thus  defined,  is  the  time  intervening 
between  two  successive  transits  of  the  same  star ;   at  least,  it  is  so 
within  the  hundredth  part  of  a  second,  though  on  account  of  the 
precession  of  the  equinoxes  (and  the  proper  motions  of  the  stars) 
the  agreement  is  not  absolute,  the  difference  amounting  to  about 
one  day  in  twenty-six  thousand  years'. 

111.  Apparent  Solar  Time.  —  Just  as  sidereal  time  is  the  hour- 
angle  of  the  vernal  equinox,  so  at  any  moment  the  apparent  solar 
time  is  the  hour-angle  of  the  sun.     It  is  the  time  shown  by  the  sun- 
dial, and  its  noon  is  when  the  sun  crosses  the  meridian.     On  account 


DETERMINATION   OF   TIME.  79 

of  the  annual  eastward  motion  of  the  sun  among  the  stars  (due  to 
the  earth's  orbital  motion),  this  day  is  about  four  minutes  longer 
than  the  sidereal ;  i.e.,  while  the  earth's  revolution  brings  our  me- 
ridian back  to  a  given  star  in  just  twenty-four  (sidereal)  hours,  it 
takes  ^^  of  a  day  longer  in  the  sun's  case.  Moreover,  because  the 
sun's  motion  in  right  ascension  is  not  uniform,  the  apparent  solar 
days  are  not  all  of  the  same  length,  nor,  consequently,  its  hours, 
minutes,  or  seconds.  December  23d  is  fifty-one  seconds  longer  from 
(apparent)  noon  to  noon  than  September  16th.  For  this  reason, 
apparent  solar  time  is  not  satisfactory  for  scientific  use,  and  has 
long  been  discarded  in  favor  of  mean  solar  time. 

112.  Mean  Solar  Time. — A  "fictitious  sun"  is  therefore  imagined, 
which  moves  uniformly  and  in  the  celestial  equator,  completing  its 
annual  course  in  exactly  the  same  time  as  that  in  which  the  actual 
sun  makes  the  circuit  of  the  ecliptic.     It  is  mean  noon  when  this 
"  fictitious  sun  "  crosses  the  meridian,  and  at  any  moment  the  hour- 
angle  of  this  "fictitious  sun  "  is  the  mean  time  for  that  moment. 

Sidereal  time  will  not  answer  for  business  purposes,  because  its  noon  (the 
transit  of  the  vernal  equinox)  occurs  at  ah1  hours  of  night  and  daylight  in 
different  seasons  of  the  year.  Apparent  solar  time  is  scientifically  unsatis- 
factory, because  of  the  variation  in  the  length  of  its  days  and  hours.  And 
yet  we  have  to  live  by  the  sun  ;  its  rising  and  setting,  daylight  and  night, 
control  our  actions.  In  mean  solar  time  we  find  a  satisfactory  compromise, 
an  invariable  time  unit,  and  still  an  agreement  with  sun-dial  time  close 
enough  for  convenience.  It  is  the  time  now  used  for  all  purposes  except  in 
certain  astronomical  work.  The  difference  between  apparent  time  and  mean 
time,  never  amounting  to  more  than  about  a  quarter  of  an  hour,  is  called 
the  equation  of  time,  and  will  be  discussed  hereafter  in  connection  with  the 
earth's  orbital  motion,  Chap.  VI. 

The  nautical  almanac  furnishes  data  by  means  of  which  the  sidereal 
time  may  be  deduced  from  the  corresponding  solar,  or  vice  versa,  by 
a  very  brief  and  simple  calculation.  See  Appendix,  Art.  1000. 

113.  In  practice  the  problem  of  determining  the  time  always, 
takes  the  form  of  ascertaining  the  error  of  a  time-piece ;  that  is,  the 
amount  by  which  a  clock  or  watch  is  fast  or  slow  of  the  time  it 
ought  to  show.     The  methods  most  in  use  by  astronomers  are  the 
following  :  — 

First.  By  means  of  the  transit  instrument.  Since  the  right  ascen- 
sion of  a  star  is  the  sidereal  time  of  its  passage  across  the  meridian 


80  PROBLEMS   OF   PRACTICAL   ASTRONOMY. 

(Art.  26) ,  it  is  obvious  that  the  difference  between  the  right  ascension 
of  a  known  star  and  the  time  shown  by  a  sidereal  clock  at  the  instant 
when  the  star  crosses  the  middle  wire  of  an  accurately  adjusted 
transit  instrument,  is  the  error  of  the  clock  at  that  moment.  Prac- 
tically, it  is  usual  to  observe  a  number  of  stars  (from  eight  to  ten) , 
reversing  the  instrument  once  at  least,  so  as  to  eliminate  the  collima- 
tion  error  (Art.  60) .  With  a  good  instrument  a  skilled  observer  can 
determine  this  clock  error  or  "correction"  within  about  one-thirtieth 
of  a  second  of  time,  provided  proper  means  are  taken  to  ascertain 
and  allow  for  his  "personal  equation." 

114.  Personal  Equation.  — It  is  found  that  every  observer  has  his 
own  peculiarities  of  time  observation  with  a  transit,  and  his  "personal 
equation "  is  the  amount  to   be   added  (algebraically)   to  the  time 
observed  by  him,  in  order  to  get  the  actual  moment  of  transit  as  it 
would  be  recorded  by  some  supposable  arrangement,  which  should 
automatically  register  the  moment  when  the  star's  image  was  bisected 
by  the  wire. 

This  personal  equation  differs  for  different  observers,  but  is  reasonably 
(though  never  strictly)  constant  for  one  who  has  had  much  practice.  In  the 
case  of  observations  with  the  chronograph,  it  is  usually  less  than  ±  08.2.  It 
can  be  determined  by  an  apparatus  in  which  an  artificial  star,  resembling 
the  real  stars  as  much  as  possible  in  appearance,  is  made  to  traverse  the  field 
of  view  and  to  telegraph  its  arrival  at  certain  wires,  while  the  observer  notes 
the  moments  for  himself. 

One  of  the  most  important  problems  of  practical  astronomy  now  awaiting 
solution  is  the  contrivance  of  some  practical  method  of  time  observation 
free  from  this  annoying  human  element.  Attempts  are  being  made  to  utilize 
photography,  and  with  fair  prospects  of  success. 

If  mean  time  is  wanted,  it  can  be  deduced  from  the  sidereal  time 
by  the  data  of  the  almanac. 

The  sun  can  also  be  observed  instead  of  the  stars,  the  moment  of 
the  sun's  transit  being  that  of  apparent  noon ;  but  this  observation, 
for  many  reasons,  is  far  less  accurate  and  satisfactory  than  observa- 
tions of  the  stars. 

115.  Second.    The  method  of  equal  altitudes.  — If  we  observe  with 
a  sextant  in  the  forenoon  the  time  shown  by  the  chronometer  when  the 
sun  attains  the  height  indicated  by  a  certain  reading  of  the  sextant,  and 
then  in  the  afternoon,  the  time  when  the  sun  again  reaches  the  same 


DETERMINATION  OF  TIME. 


81 


altitude,  the  moment  of  apparent  moon  will  be  half-way  between  the 
two  observed  times  ;  provided,  of  course,  that  the  chronometer  runs 
uniformly  during  the  interval,  and  also  provided  that  proper  correc- 
tion is  made  for  the  sun's  slight  motion  in  declination  —  a  correction 
easily  computed. 

The  advantage  of  this  method  is  that  the  errors  of  graduation  of 
the  sextant  have  no  effect,  nor  is  it  necessary  for  the  observer  to 
know  his  latitude  except  approximately. 

Per  contra,  there  is,  of  course,  danger  that  the  afternoon  observa- 
tions may  be  interfered  with  by  clouds  ;  and,  moreover,  both  obser- 
vations must  be  made  at  the  same  place. 

A  modification  of  this  method  is  now  coming  into  extensive  use, 
in  which  two  different  stars  of  known  right  ascension  and  of  nearly 
the  same  declination  are  used,  at  equal  altitudes  east  and  west  of 
the  meridian. 

116.  Third.  By  a  single  altitude  of  the  sun,  the  observer's  latitude 
being  known.  —  This  is  the  method  usual  at  sea.  The  altitude  of  the  sun 
having  been  measured  with  the  sextant,  and  the  corresponding  time 
shown  by  the  chronometer  having  been  accurately  noted,  we  compute 

the  hour-angle  of  the  sun,  P, 
from  the  triangle  ZPS  (Fig. 
41),  and  this  hour-angle  cor- 
rected for  the  equation  of 
time,  gives  the  true  mean 
time  at  the  observed  moment. 
The  difference  between  this 
and  that  shown  by  the  chro- 
nometer is  the  error  of  the 

FIG.  41.—  Determination  of  Time  by  a  Single  Altitude.  / 

chronometer.    In  the  triangle 

ZPS  all  three  of  the  sides  are  given,  viz.  :  PZ  is  the  complement 
of  the  latitude  <£,  which  is  supposed  to  be  known  ;  PS  is  the  com- 
plement of  the  declination  8,  which  is  found  in  the  almanac,  as  is 
also  the  equation  of  time  ;  while  ZS,  or  £,  is  the  complement  of  the 
sun's  altitude,  as  measured  by  the  sextant,  and  corrected  for  semi- 
diameter,  refraction,  and  parallax.  The  formula  is 


EH 


sin  i  P  =  /si 
\ 


~  S)]  sin  j  [g  -  (<fr  -  8)] 


cos 


cos  8 


In  order  to  accuracy,  it  is  desirable  that  the  sun  should  be  on  the 
prime  vertical,  or  as  near  it  as  practicable.   It  should  not  be  near  the 


82  PKOBLEMS    OF    PRACTICAL   ASTKONOMY. 

meridian.  Any  slight  error  in  the  assumed  latitude  produces  no 
sensible  effect  upon  the  result,  if  the  sun  is  exactly  east  or  west  at 
the  time  the  observation  is  taken.  The  disadvantage  of  the  method 
is  that  any  error  of  graduation  of  the  sextant  vitiates  the  result. 

In  some  cases  a  person  is  so  situated  that  it  is  necessary  to  determine  his 
time  roughly,  without  instruments ;  and  this  can  be  done  within  about  a 
half  a  minute  by  establishing  a  noon-mark,  which  is  nothing  but  a  line 
drawn  exactly  north  and  south,  with  a  plumb-line,  or  some  vertical  edge, 
like  the  edge  of  a  door-frame  or  window  sash,  at  its  southern  extremity. 
The  shadow  will  then  always  fall  upon  the  meridian  line  at  apparent  noon. 

117.  The  Civil  and  the  Astronomical  Day.  —  The  astronomical  day 
begins  at  mean  noon.1     The  civil  day  begins  at  midnight,  twelve 
hours  earlier.     Astronomical  mean  time  is  reckoned  round  through 
the  whole  twenty-four  hours,  instead  of  being  counted  in  two  series 
of  twelve  hours  each.     Thus,  10  A.M.  of  Wednesday,  May  2,  civil 

reckoning,  is  Tuesday,  May  1,  22h,  by  astronomical  reckoning. 

t 

LONGITUDE. 

118.  Having  now  methods  of  obtaining  the  true  local  time,  we 
can  attack  the  problem  of  longitude,  which  is  perhaps  the  most 
important  of  all  the  economic  problems  of  astronomy.     The  great 
observatories  at  Greenwich  and  at  Paris  were  established  simply  for 
the  purpose  of  furnishing  the  observations  which  could  be  made  the 
basis  of  the  accurate  determination  of  longitude  at  sea. 

The  longitude  of  a  place  on  the  earth  is  the  angle  at  the  pole  between 
the  meridian  of  Greenwich  and  the  meridian  passing  through  the  ob- 
server's place ;  or  it  is  the  arc  of  the  equator  intercepted  between 
these  meridians  ;  or,  what  comes  to  the  same  thing,  since  this  arc  is 
measured  by  the  time  required  for  the  earth  to  turn  sufficiently  to 
bring  the  second  meridian  into  the  same  position  held  by  the  first, 
it  is  simply  the  difference  of  their  local  times,  —  the  amount  by  which 
the  noon  at  Greenwich  is  earlier  or  later  than  at  the  observer's  place. 
It  is  now  usually  reckoned  in  hours,  minutes,  and  seconds,  instead 
of  degrees. 

Since  it  is  easy  for  the  observer  to  find  his  own  local  time  by  the 
methods  which  have  been  given,  the  knot  of  the  problem  is  really 
this  :  being  at  any  place,  to  find  the  corresponding  local  time  at  Green- 
wich without  going  there. 
» 

1  There  is  a  proposition  to  make  the  astronomical  day  correspond  with  the 
civil,  which  has  met  with  some  favor.  But  practical  astronomers  dislike  to  have 
to  change  dates  at  midnight,  in  the  midst  of  their  work. 


LONGITUDE.  83 

The  methods  of  finding  the  longitude  may  be  classed  under  three 
different  heads : 

First,  By  means  of  signals  simultaneously  observable  at  the  places 
between  which  the  difference  of  longitude  is  to  be  found. 

Second,  By  making  use  of  the  moon  as  a  clock-hand  in  the  sky. 

Third,  By  purely  mechanical  means,  such  as  chronometers  and 
the  telegraph.  This  is  the  modern  method,  and  the  best  wherever 
available. 

119.  Under  the  first  head  we  may  make  use  of 

[A]  A  Lunar  ^Eclipse.  —  When  the  moon  enters  the  shadow  of 
the  earth,  the  phenomenon  is  seen  at  the  same  moment,  no  matter 
where  the  observer  may  be.     By  noting,  therefore,  his  own  local 
time  at  the  moment,  and  afterwards  comparing  it  with  the  time  at 
which  the  phenomenon  was  observed  at  Greenwich,  he  will  obtain  his 
longitude  from  Greenwich.     Unfortunately,  the  edge  of  the  earth's 
shadow  is  so  indistinct  that  the  progress  of  events  is  very  gradual, 
so  that  sharp  observations  are  impossible. 

[B]  ^Eclipses  of  the  satellites  of  Jupiter  may  be  used  in  the  same 
way,  with  the  advantage  that  they  occur  very  frequently,  —  almost 
every  night,  in  fact ;  but  the  objection  to  them  is  the  same  as  to  the 
lunar  eclipses,  —  they  are  not  sudden. 

[C]  The  appearance  and  disappearance  of  meteors  may  be  and  has 
been  used  to  determine  the  difference  of  longitude  between  places 
not  more  than  two  or  three  hundred  miles  apart,  and  gives  very 
accurate  results.     (Now  superseded  by  the  telegraph.) 

[D]  Artificial  signals,  such  as  flashes  of  powder  and  rockets,  can 
be  used  between  two  stations  not  too  far  distant.     Early  in  the  cen- 
tury the  difference  of  longitude  between  the  Black  Sea  and  the 
Atlantic  was  determined  by  means  of  a  chain  of  signal  stations  on 
the  mountain  tops  ;  so  also,  later,  the  difference  of  longitude  between 
the  eastern  and  western  extremities  of  the  northern  boundary  of 
Mexico.     This  method  is  now  superseded  by  the  telegraph. 

120.  SECOND,  the  moon  regarded  as  a  clock. 

Since  the  moon  revolves  around  the  earth  once  a  month,  it  is,  of 
course,  continually  changing  its  place  among  the  stars ;  and  as  the 
laws  of  its  motion  are  now  well  known,  and  as  the  place  which  it 
will  occupy  is  predicted  for  every  hour  of  every  Greenwich  day 
three  years  in  advance  in  the  nautical  almanac,  it  is  possible  to  de- 
duce the  corresponding  Greenwich  time  by  any  observation  which 
will  determine  the  place  of  the  moon  among  the  stars!  The  almanac 


84  PROBLEMS   OF   PRACTICAL  ASTRONOMY. 

place,  however,  is  the  place  at  which  the  moon  would  be  seen  by  an 
observer  at  the  centre  of  the  earth,  and  consequently  the  actual  ob- 
servations are  in  most   cases   complicated   with   very   disagreeable 
reductions  for  parallax  before  they  can  be  made  available. 
The  simplest  lunar  method  is, 

[A]  That  of  Moon  Culminations.  —  We   merely  observe  with  a 
transit  instrument  the  time  when  the  moon's  bright  limb  crosses  the 
meridian  of   the  place ;  and  immediately  after  the  moon  we  observe 
one  or  more  stars  with   the  same  instrument,  to  give  us  the  error 
of  our  clock.     As  the  moon  is  observed  on  the  meridian,  its  paral- 
lax does  not  affect  its  right  ascension,  and  accordingly,  by  a  simple 
reference  to  the  almanac,  we  can  ascertain  the  Greenwich  time  at 
which  the  moon  had  the  particular  right  ascension  determined  by 
the  observation.     The  method  has  been  very  extensively  used,  and 
would  be  an  admirable  one  were  it  not  for  the  effects  of  personal 
equation. 

It  seldom  happens  that  the  personal  equation  of  an  observer  is  the  same 
for  such  an  object  as  the  limb  of  the  moon  as  it  is  for  a  star ;  and  since  the 
moon's  motion  among  the  stars  is  very  slow,  the  effect  of  such  a  difference 
is  multiplied  by  about  30  (roughly  the  number  of  days  in  a  month)  in  its 
effect  upon  the  longitude  deduced. 

[B]  Lunar- Distances,,  —  At   sea   it   is,   of   course,  impossible   to 
observe  the  moon  with  a  transit  instrument,  but  we  can  observe  its 
distance  from  the  stars  near  its  path  by  means  of  a  sextant.     The 
distance    observed  will  not  be   the   same   that  it  would   be   if   the 
observer  were   at  the  centre  of   the   earth,  but   by  a   mathematical 
process   called  ' '  clearing  a  lunar "  the   distance   as   seen  from   the 
centre  of  the  earth  can  be  easily  deduced,  and  compared  with  the 
distance   given   in    the   almanac.     From   this   the  longitude  can  be 
determined.      Any   error,  however,   in   measuring   a   lunar-distance 
entails  an  error  about  thirty  times  as  great  in  the  resulting  longitude, 
and  the  method  is  at  present  very  little  used,  the  moon  having  been 
superseded  by  the  chronometer  for  such  purposes. 

[C]  Occultations.  —  Occasionally,  in  its  passage  through  the  sky, 
the  moon  over-runs  a  star,  or  "  occults"  it.     The  star  vanishes  instan- 
taneously, and,  of  course,  at  the  moment  of  its  disappearance  the 
distance  from  the  centre  of  the  moon  to  the  star  is  precisely  equal 
to  the  apparent  semi-diameter  of  the  moon  ;  we  thus  have  a  "  lunar- 
distance  "  self-measured. 

Observations  of  this  kind  furnish  one  of  the  most  accurate  methods 


LONGITUDE.  85 

of  determining  the  difference  of  longitude  between  widely  separated 
places,  the  only  difficulty  arising  from  the  fact  that  the  edge  of  the 
moon  is  not  smooth,  but  more  or  less  mountainous,  so  that  the  dis- 
tance of  a  star  from  the  moon's  centre  is  not  always  the  same  at 
the  moment  of  its  disappearance, 

[D]  In  the  same  way  a  solar  eclipse  may  be  employed  by  observing 
the  momejit  when  the  moon's  limb  touches  that  of  the  sun. 

It  will  be  noticed  that  these  two  last  methods  (the  methods  of  occupation 
and  solar  eclipse)  do  not  belong  in  the  same  class  with  the  method  of  lunar 
eclipse,  because  the  phenomena  are  not  seen  at  the  same  instant  at  different 
places,  but  the  calculation  of  longitude  depends  upon  the  determination  of 
the  moon's  place  in  the  sky  at  the  given  time,  as  seen  from  the  earth's 
centre. 

There  are  still  other  methods,  depending  upon  measurements  of 
the  moon's  position  by  observations  of  its  altitude  or  azimuth.  In 
all  such  cases,  however,  every  error  of  observation  entails  a  vastly 
greater  error  in  the  final  results.  Lunar  methods  (excepting  occul- 
tations)  are  only  used  when  better  ones  are  unavailable. 

121.  Finally  we  have  what  may  be  called  the  mechanical  methods 
of  determining  the  longitude. 

[A]  By  the  chronometer;  which  is  simply  an  accurate  watch  that 
has  been  set  to  indicate  Greenwich  time  before  the  ship  leaves  port. 
In  order  to  find  the  longitude  by  the  chronometer,  the  sailor  has  to 
determine  its  "error"  upon  local  time  by  an  observation  of  the  alti- 
tude of  the  sun  when  near  the  prime  vertical,  as  indicated  on  page  78. 
If  the  chronometer  indicates  true  Greenwich  time,  the  error  deduced 
from  the  observation  will  be  the  longitude.  Usually,  however,  the  indi- 
cation of  the  chronometer  face  requires  correction  for  the  rate  and 
run  of  the  chronometer  since  leaving  port. 

Chronometers  are  only  imperfect  instruments,  and  it  is  important,  there- 
fore, that  several  of  them  should  be  used  to  check  each  other.  It  requires 
three  at  least,  because  if  only  two  chronometers  are  carried  and  they  disagree, 
there  is  nothing  to  indicate  which  one  is  the  delinquent. 

On  very  long  voyages  the  errors  of  chronometers  are  cumulative,  and  the 
uncertainty  accumulates  much  more  rapidly  than  in  proportion  to  the  time  ; 
i.e.,  if  the  error  to  be  feared  in  the  use  of  a  chronometer  in  longitude  deter- 
minations at  the  end  of  a  week  is  about  two  seconds  of  time,  at  the  end  of 
the  month  it  would  be,  not  eight  seconds,  but  very  likely  twenty  or  thirty, 
owing  to  the  possible  changes  of  its  rate  during  the  voyage. 

If,  therefore,  a  ship  is  to  be  at  sea,  without  making  port,  more  than  three 


86  PROBLEMS   OF   PRACTICAL   ASTRONOMY. 

or  four  months  at  a  time,  the  method  becomes  untrustworthy,  and  it  may  be 
necessary  to  recur  to  lunar  distances ;  for  voyages  of  less  than  a  month  the 
method  is  now,  practically,  all  that  could  be  desired. 

[B]  But  the  method  which,  wherever  it  is  applicable,  has  super- 
seded all  others,  is  that  of  The  Telegraph.  When  we  wish  to  find  the 
longitude  between  two  stations  connected  by  telegraph,  the  process 
is  usually  as  follows  :  The  observers  at  both  stations,  after  ascer- 
taining that  they  both  have  clear  weather,  proceed  to  determine  their 
own  local  time  by  extensive  series  of  star  observations  with  the 
transit  instrument.  Then,  at  an  agreed-upon  time,  the  observer  at 
Station  A  "switches  his  clock"  into  the  telegraphic  circuit,  so  that 
its  beats  are  communicated  along  the  line  and  received  upon  the  chron- 
ograph of  the  other,  say  the  western  station.  After  the  eastern  clock 
has  thus  sent  its  signals,  say  for  two  minutes,  it  is  switched  out  of 
the  circuit,  and  the  western  observer  now  switches  his  clock  into  the 
circuit,  and  its  beats  are  received  upon  the  eastern  chronograph.  The 
operation  is  closed  b}'  another  series  of  star  observations. 

We  have  now  upon  each  chronograph  sheet  an  accurate  comparison 
of  the  two  clocks,  showing  the  amount  by  which  the  western  clock  is 
slow  of  the  eastern.  If  the  transmission  of  electric  signals  were 
instantaneous,  the  difference  shown  upon  the  two  chronograph  sheets 
would  agree  precisely.  Practically,  however,  there  will  always  be  a 
small  discrepancy  amounting  to  twice  the  time  occupied  in  the  trans- 
mission of  the  signals ;  but  the  mean  of  the  two  differences  will  be 
the  true  difference  of  longitude  of  the  places  after  the  proper  correc- 
tions have  been  applied.  Especial  care  must  be  taken  to  determine 
with  accuracy,  or  to  eliminate,  the  personal  equations  of  the  observers. 

It  is  customary  to  make  observations  of  this  kind  on  not  less  than  five 
or  six  evenings  in  cases  where  it  is  necessary  to  determine  the  difference  of 
longitude  with  the  highest  accuracy.  The  astronomical  difference  of  longi- 
tude between  two  places  can  thus  be  telegraphically  determined  within  about 
the  one-hundredth  part  of  a  second  of  time ;  i.e.,  within  about  ten  feet  or  so, 
in  the  latitude  of  the  United  States. 

It  may  be  noted  here  that  the  time  occupied  by  the  transmission  of  elec- 
tric signals  in  longitude  operations  is  not  to  be  taken  as  the  real  measure  of 
"  the  velocity  of  the  electric  fluid  "  upon  the  wires,  as  was  once  supposed. 
The  time  apparently  consumed  in  the  transmission  is  simply  the  time  re- 
quired for  the  current  at  the  receiving  station  (which  current  probably 
begins  at  the  very  instant  the  key  is  touched  at  the  other  end  of  the  line)  to 
become  strong  enough  to  do  its  work  in  making  the  signal ;  and  this  time 
depends  upon  a  multitude  of  circumstances. 


LONGITUDE.  87 

122.  Local  and  Standard  Time, — In  connection  with  time  and 
longitude  determinations,  a  few  words  on  this  subject  will  be  in  place.  Un- 
til recently  it  has  always  been  customary  to  use  only  local  time,  each  observer 
determining  his  own  time  by  his  own  observations.  Before  the  days  of  the 
telegraph,  and  while  travel  was  comparatively  slow  and  infrequent,  this  was 
best ;  but  the  telegraph  and  railway  have  made  such  changes  that,  for  many 
reasons,  it  is  better  to  give  up  the  old  system  of  local  times  in  favor  of  a 
system  of  standard  time.  It  facilitates  all  railway  and  telegraphic  busi- 
ness in  a  remarkable  degree,  and  makes  it  practically  easy  for  every  one  to 
keep  accurate  time,  since  it  can  be  daily  wired  from  some  observatory  to 
every  telegraph  office. 

According  to  the  system  that  is  now  established  in  this  country,  there  are 
five  such  standard  times  in  use,  —  the  colonial,  the  eastern,  the  central,  the 
mountain,  and  the  Pacific,  —  which  differ  from  Greenwich  time  by  exactly 
four,  five,  six,  seven,  and  eight  hours  respectively,  the  minutes  and  seconds  being 
identical  everywhere.  At  most  places  only  one  of  these  times  is  employed ; 
but  in  cities  where  different  systems  join  each  other,  there  are  two  standard 
times  in  use,  differing  from  each  other  by  exactly  one  hour,  and  from  the 
local  time  by  about  half  an  hour.  In  some  such  places  the  local  time  also 
maintains  its  place. 

In  order  to  determine  the  standard  time  by  observation,  it  is  only  nec- 
essary to  determine  the  local  time  by  one  of  the  methods  given,  and  correct 
it  according  to  the  observer's  longitude  from  Greenwich. 


123.  Where  the  Day  Begins.  —  If  we  imagine  a  traveller  starting 
from  Greenwich  on  Monday  noon,  and  journeying  westward  as  swiftly  as  the 
earth  turns  to  the  east  under  his  feet,  he  would,  of  course,  keep  the  sun  exactly 
on  the  meridian  all  day  long,  and  have  continual  noon.  But  what  noon  ? 
It  was  Monday  when  he  started,  and  when  he  gets  back  to  London,  twenty- 
four  hours  later,  it  is  Tuesday  noon  there,  and  there  has  been  no  intervening 
sunset.  When  does  Monday  noon  become  Tuesday  noon?  The  conven- 
tion is  that  the  change  of  date  occurs  at  the  ISQtk  meridian  from  Greenwich. 
Ships  crossing  this  line  from  the  east  skip  one  day  in  so  doing.  If  it  is 
Monday  forenoon  when  the  ship  reaches  the  line,  it  becomes  Tuesday  fore- 
noon the  moment  it  passes  it,  the  intervening  twenty-four  hours  being 
dropped  from  the  reckoning  on  the  log-book.  Vice  versa,  when  a  vessel 
crosses  the  line  from  the  western  side,  it  counts  the  same  day  twice,  passing 
from  Tuesday  forenoon  back  to  Monday,  and  having  to  do  its  Tuesday  over 
again. 

This  180th  meridian  passes  mainly  over  the  ocean,  hardly  touching  land 
anywhere.  There  is  a  little  irregularity  in  the  date  upon  the  different 
islands  near  this  line.  Those  which  received  their  earliest  European  inhabi- 
tants via  the  Cape  of  Good  Hope  have,  for  the  most  part,  the  Asiatic  date, 
belonging  to  the  west  side  of  the  180th  meridian ;  while  those  that  were  ap- 
proached via  Cape  Horn  have  the  American  date. 


88  PROBLEMS    OF   PRACTICAL   ASTRONOMY. 

When  Alaska  was  transferred  from  Russia  to  the  United  States,  it  was 
necessary  to  drop  one  day  of  the  week  from  the  official  dates. 

THE   PLACE   OF   A   SHIP   AT  SEA. 

124.  The  determination  of  the  place  of  a  ship  at  sea  is  commer- 
ciall}"  of  such  importance  that,  at  the  risk  of  a  little  repetition,  we 
collect  together  here  the  different  methods  available  for  its  determi- 
nation.    The  methods  employed  are  necessarily  such  that  observa- 
tions  can   be   made  with    the    sextant   and   chronometer,    the   only 
instruments  available  under  the  circumstances. 

The  Latitude  is  usually  obtained  by  observations  of  the  sun's 
altitude  at  noon,  according  to  the  method  explained  in  Art.  103. 

The  Longitude  is  usually  found  by  determining  the  error  upon  local 
time  of  the  chronometer,  which  carries  Greenwich  time.  The  nec- 
essary observations  of  the  sun's  altitude  should  be  made  when  the 
sun  is  near  the  prime  vertical,  as  explained  in  Art.  116. 

In  the  case  of  long  voyages,  or  when  the  chronometer  has  for  any 
reason  failed,  the  longitude  may  also  be  obtained  by  measuring  a 
lunar-distance  and  comparing  it  with  the  data  of  the  nautical  almanac. 

By  these  methods  separate  observations  are  necessary  for  the  lati- 
tude and  for  the  longitude. 

125.  Sumner's   Method.  —  Recently  a  new  method,  first  proposed 
by  Captain  Sumner,  of  Boston,  in  1843,  has  been  coming  largely  into 
use.     In  this  method,  each  observation  of  the  sun's  altitude,  with  the 
corresponding  chronometer  time,  is  made  to  define  the  position  of  the 
ship  upon  a  certain  line,  called  the  circle  of  position.     Two  such  ob- 
servations will,  of  course,  determine  the  exact  place  of  the  vessel  at 
one  of  the  intersections  of  the  two  circles. 

At  any  moment  the  sun  is  vertically  over  some  point  upon  the 
earth's  surface,  which  maybe  called  the  sub-solar  point.  An  observer 
there  would  have  the  sun  directly  overhead.  Moreover,  if  at  any 
point  on  the  earth  an  observer  measures  the  altitude  of  the  sun  with 
his  sextant,  the  zenith  distance  of  the  sun  (which  is  the  complement 
of  this  altitude)  will  be  his  distance  from  the  sub-solar  point  at  the 
moment  of  observation,  reckoned  in  degrees  of  a  great  circle. 

If,  then,  I  take  a  terrestrial  globe,  and,  opening  the  dividers  so  as 
to  cover  an  arc  equal  to  this  observed  zenith  distance  of  the  sun, 
put  one  foot  of  the  dividers  upon  the  sub-solar  point,  and  sweep  a 


POSITION    AT    SEA.  89 

circle  on  the  surface  of  the  globe  around  that  point,  the  observer 
must  "be  somewhere  on  the  circumference  of  that  circle;  and  moreover, 
if  to  the  observer  the  sun  is  in  the  southwest,  he  himself  must  be  in 
the  opposite  direction  from  this  sub-solar  point;  i.e.,  northeast  of  it. 
In  other  words,  the  azimuth  of  the  sun  at  the  time  of  observation 
informs  him  upon  what  part  of  the  circle  he  is  situated. 

Suppose  a  similar  observation  made  at  the  same  place  a  few  hours 
later.  The  sub-solar  point,  and  the  zenith  distance  of  the  sun,  will 
have  changed ;  and  we  shall  obtain  a  new  circle  of  position,  with  its 
centre  at  the  new  sub-solar  point.  The  observer  must  be  at  one  of 
its  two  intersections  with  the  first  circle  —  which  of  the  two  inter- 
sections is  easily  determined  from  the  roughly  observed  azimuth  of 
the  sun. 

If  the  ship  moves  between  the  two  observations,  the  proper  allow- 
ance must  be  made  for  the  motion.  This  is  easily  done  by  shift- 
ing upon  the  chart  that  part  of  the  first  circle  of  position  where  the 
ship  was  situated,  carrying  the  line  forward  parallel  to  itself,  by  an 
amount  just  equal  to  the  ship's  run  between  the  two  observations, 
as  shown  by  the  log.  The  intersection  with  the  second  circle  then 
gives  the  ship's  place  at  the  time  of  the  second  observation. 

The  only  problem  remaining  is  to  find  the  position  of  the  "  sub-solar 
point "  at  any  given  moment.  Now,  the  latitude  of  this  point  is  ob- 
viously the  declination  of  the  sun  (which  is  found  in  the  almanac). 
If  the  sun's  declination  is  zero,  the  sun  is  vertically  over  some  point 
upon  the  equator.  If  its  declination  is  -f-  20°,  it  is  vertically  over 
some  point  on  the  twentieth  parallel  of  north  latitude,  etc. 

In  the  next  place,  its  longitude  is  equal  to  the  Greenwich  apparent 
solar  time  at  the  moment  of  observation;  and  this  is  given  by  the 
chronometer  (which  keeps  Greenwich  mean  solar  time) ,  by  simply  add- 
ing or  subtracting  the  equation  of  time  ;  so  that,  by  looking  in  his 
almanac  and  at  his  chronometer,  the  observer  has  the  position  of  the 
sub-solar  point  immediately  given  him. 

Suppose,  for  example,  that  on  May  20  (the  sun's  declination  being  +  20°), 
at  11  A.M.,  Greenwich  apparent  time  (i.e.,  May  19,  23h  by  astronomical  reck- 
oning), according  to  the  chronometer,  the  sun  is  observed  to  have  an  altitude 
of  40°  by  a  ship  in  the  North  Atlantic.  -The  sub-solar  point  will  then  be 
(Fig.  42)  at  a  point  in  Africa  having  a  latitude  of  +  20°,  and  an  east  longi- 
tude of  15° —  at  A  in  the  figure.  And  the  radius  of  the  "  circle  of  position," 
i.e.,  the  distance  from  A  to  C — will  be  50°. 

Again,  a  second  observation  is  made  three  hours  later,  when  the  sun's 
altitude  is  found  to  be  65°.  The  sub-solar  point  will  then  be  at  B,  latitude 

9        ° 


90 


PEOBLEMS    OF   PRACTICAL   ASTRONOMY. 


FlQ,  42.  —  Sumner's  Method. 


20°,  longitude  30°  W.,  and  the  radius  of  the  circle  of  position  BC  will  be 
25°,  C  being  the  ship's  place. 

126.  It  is,  however,  seldom  convenient  to  carry  a  large  globe, 
and  in  practice  the  usual  procedure  is  the  following.     The  latitude 
of  a  ship  is   always   known 

within  a  few  degrees  by  the 
"  dead-reckoning  "  ;  suppose 
that  it  is  known  to  be  about 
50°  30'.  From  the  first  ob- 
servation calculate  (by  the 
methods  of  Art.  121)  what  the 
longitude  would  be  if  the  lati- 
tude were  50°,  and  also  if  it 
were  51°.  Mark  the  two 
points  on  the  charts  and  con- 
nect them  by  a  straight  line, 
which  will  be  (very  nearly)  a 
portion  of  the  first  circle  of 
position.  In  the  same  way 
obtain  a  second  "position 
line7'  from, the  second  observation.  The  intersection  of  the  two 
lines  will  give  the  ship's  place,  the  first  position  line  being  moved 
forward,  parallel  to  itself,  by  the  amount  of  the  ship's  motion  in  the 
interval  between  the  two  observations. 

The  peculiar  advantage  of  the  method  is,  that  a  single  observation 
is  used  for  all  it  is  worth,  giving  accurately  the  position  of  a  line 
upon  which  the  ship  is  somewhere  situated,  and  approximately  (by 
the  rough  observation  of  the  sun's  azimuth)  the  part  of  that  line 
upon  which  its  place  will  be  found.  In  approaching  the  American 
coast,  for  instance,  if  an  observation  be  taken  in  the  forenoon,  the 
ship's  position  circle  will  lie  nearly  parallel  to  the  coast,  and  then  a 
single  observation  will  give  approximately  the  distance  of  the  ship 
from  land,  which  may  be  all  the  sailor  wishes  to  know.  The  obser- 
vations need  not  be  taken  at  any  particular  time.  We  are  not  limited 
to  observations  at  noon,  or  to  the  time  when  the  sun  is  near  the  prime 
vertical.  It  is  to  be  noted,  however,  that  everything  depends  upon 
the  chronometer,  as  much  as  in  the  ordinary  chronometric  determi- 
nation of  longitude. 

127.  Determination  of  Azimuth.  —  A  problem,  important,  though 
not  so  often  encountered  as  that  of  latitude  and  longitude  determina- 


PROBLEMS    OF    PRACTICAL   ASTRONOMY. 


91 


tions,  is  that  of  determining  the  azimuth, or  true  bearing,  of  a  line  upon 
the  earth^s  surface.  The  process  is  this  :  With  a  theodolite  having 
an  accurately  graduated  horizontal  circle  the  observer  points  alter- 
nately upon  the  pole  star  and  upon  a  dis- 
tant signal  erected  for  the  purpose ;  the 
signal  being  an  artificial  star  consisting  of 
a  small  hole  in  a  plate  of  metal,  with  a 
bull's-eye  lantern  or  other  light  behind  it. 
It  is  desirable  that  it  should  be  at  least 
a  mile  away  from  the  observer,  so  that 
any  small  displacement  of  the  instrument 
will  be  harmless.  The  theodolite  must 
be  carefully  adjusted  for  colliniation,  and 
_  especial  pains  must  be  taken  to  have  the 

Fia.43.-DetefminfiofofAzimuth.  axis  of  the  telescope  perfectly  level. 

The  next  morning  by  daylight  the  ob- 
server measures  the  angle  or  angles  between  the  night-signal  and 
the  objects  whose  azimuth  is  required. 

If  the  pole  star  were  exactly  at  the  pole,  the  mere  difference 
between  the  two  readings  of  the  circle,  obtained  when  the  telescope 
is. pointed  on  the  star  and  on  the  signal,  would  directly  give  the 
azimuth  of  the  signal.  As  this  is  not  the  case,  however,  the  time 
at  which  each  observation  of  the  pole  star  is  made  must  be  noted, 
and  the  azimuth  of  the  star  must  be  computed  for  that  moment. 
This  can  easily  be  done,  as  the  right  ascension  and  declination  of 
this  star  are  given  in  the  almanac  for  every  day  of  the  year. 


Recurring  to  the  Z.P.S.  [zenith-pole-star]  triangle,  JV(Fig.43)  being  the 
north  point  of  the  horizon,  P  the  pole,  and  NZ  the  meridian,  we  at  once 
see  that  the  side  PS  is  the  complement  of  the  star's  declination  ;  the  side 
PZ  is  the  complement  of  the  observer's  latitude  (which  must  be  known) ; 
and  the  angle  at  P  is  the  difference  between  the  right  ascension  of  the  pole 
star  and  the  sidereal  time  of  the  observation  ;  \_(t  —  a)  if  the  star  is  west  of 
the  meridian  at  the  time,  and  (a  —  £)  if  it  is  east.]  This  will  come  out  in 
hours,  of  course,  and  must  be  reduced  to  degrees  before  making  the  com- 
putation. We  thus  have  two  sides  of  the  triangle,  viz.,  PS  and  PZ,  with 
the  included  angle  at  P,  from  which  to  compute  the  angle  Z  at  the  zenith. 
This  is  the  star's  azimuth. 

The  pole  star  is  used  because,  being  so  near  the  pole,  any  slight  error  in 
the  assumed  latitude  of  the  place  or  in  the  sidereal  time  of  the  observation 
will  hardly  produce  any  effect  upon  the  result,  especially  if  the  star  be 
caught  between  five  and  six  hours  before  or  after  its  upper  culmination,  at 


92  POSITION  OF  A  HEAVENLY   BODY. 

a  time  when  it  changes  its  azimuth  very  slowly  (near  Sr  or  S"  in  the  figure). 
The  sun,  or  any  other  heavenly  body  whose  position  is  given  in  the  almanac, 
can  also  be  used  as  a  reference  point  in  the  same  way,  provided  sufficient 
pains  is  taken  to  secure  an  accurate  observation  of  the  time  at  the  instant 
when  the  pointing  is  made.  The  altitude  should  not  exceed  thirty  degrees 
or  so.  But  the  results  are  usually  rough  compared  with  those  obtained  by 
means  of  the  pole  star. 

DETERMINATION  OF  THE  POSITION  OF  A  HEAVENLY  BODY. 

128.  The   position   of   a  heavenly  body  is   defined  by  its  right 
ascension  and  declination.     These  quantities  may  be  determined  — 

(1)  By  the  meridian  circle,  provided  the  body  is  bright  enough  to 
be  seen  by  the  instrument  and  comes  to  the  meridian  in  the  night- 
time. If  the  instrument  is  in  exact  adjustment,  the  sidereal  time 
when  the  object  crosses  the  middle  wire  of  the  reticle  of  the  instrument  is 
directly  (according  to  Art.  27)  the  right  ascension  of  the  object. 

The  reading  of  the  circle  of  the  instrument,  corrected  for  refraction 
and  parallax  if  necessary,  gives  the  polar  distance  of  the  object,  if 
the  polar  point  of  the  circle  has  been  determined  (Art.  66)  ;  or  it  gives 
the  zenith  distance  of  the  object  if  the  nadir  point  has  been  deter- 
mined (Art.  67).  In  either  case  the  declination  can  be  immediately 
deduced,  being  the  complement  of  the  polar  distance,  and  equal  to 
the  latitude  of  the  observer,  minus  the  distance  of  the  star  south  of 
the  zenith.  One  complete  observation,  then,  with  the  meridian  circle, 
determines  both  the  right  ascension  and  declination  of  the  object. 

It  is  often  better  to  use  the  instrument  "  differentially,"  i.e.,  to  observe 
some  standard  star,  whose  place  is  already  accurately  known,  along  with  the 
object  whose  place  is  to  be  determined.  We  thus  obtain  the  difference  of 
their  Right  Ascensions  and  Declinations,  and  slight  errors  in  the  graduation 
and  adjustment  of  the  instrument  affect  the  final  result  far  less  than  in  an 
"  absolute  "  determination. 

If  a  body  (a  comet,  for  instance)  is  too  faint  to  be  observed  by 
the  telescope  of  the  meridian  circle,  which  is  seldom  very  powerful, 
or  if  it  does  not  come  to  the  meridian  during  the  night,  we  usually 
accomplish  our  object  — 

129.  (2)    By  the  Equatorial,  determining  the  position  of  the  body 
by  measuring   the   difference  of  right  ascension  and  declination  be- 
tween it  and  some  neighboring  star,  whose  place  is  given  in  a  star 
catalogue,  and  of  course  has  been  determined  by  the  meridian  circle 
of  some  observatory. 


POSITION    OF   A   HEAVENLY    BODY.  93 

In  measuring  this  difference  of  right  ascension  and  declination,  we  usually 
employ  a  filar  micrometer  fitted  like  the  reticle  of  a  meridian  circle.  It  car- 
ries a  number  of  wires  which  lie  north  and  south  in  the  field  of  view,  and 
these  are  crossed  at  right  angles  by  one  or  more  wires  which  can  be  moved 
by  the  micrometer  screw.  The  difference  of  right  ascension  between  the  star 
and  the  object  to  be  determined  is  measured  by  simply  observing  with  the 
chronograph  the  transits  of  the  two  objects  across  the  north  and  south 
wires  ;  the  difference  of  declination,  by  bisecting  each  object  with  one  of  the 
micrometer  wires  as  it  crosses  the  middle  of  the  field  of  view.  The  ob- 
served difference  must  be  corrected  for  refraction  and  for  the  motion  of 
the  body,  if  it  is  appreciable. 

Other  less  complicated  micrometers  are  also  in  use.  One  of  them,  called 
the  ring  micrometer,  consists  merely  of  an  opaque  ring  supported  in  the  field 
of  view  either  by  being  cemented  to  a  glass  plate  or  by  slender  arms  of 
metal.  The  observations  are  made  by  noting  the  transits  of  the  comparison 
star  and  of  the  object  to  be  determined  across  the  outer  and  inner  edges  of 
the  ring.  If  the  radius  of  the  ring  is  known  in  seconds  of  arc,  we  can 
from  these  observations  deduce  the  differences  both  of  right  ascension  and 
declination.  The  results  are  less  accurate  than  those  given  by  the  wire 
micrometer,  but  the  ring  micrometer  has  the  advantage  that  it  can  be  used 
with  any  telescope,  whether  equatorially  mounted  or  not,  and  requires  no 
adjustment. 

There  are  also  many  other  methods  of  effecting  the  same  object. 

130.  To  Compute  the  Time  of  Sunrise  or  Sunset.  —  To  solve  this  prob- 
lem, it  is  only  necessary  to  work  out  the  Z.P.S.  triangle  and  find  the  hour-angle 
P,  having  given  precisely  the  same  data  as  in  finding  the  time  by  a  single 
altitude  of  the  sun  (Art.  116).  PZ  is  the  observer's  co-latitude,  PS  is  the 
complement  of  the  sun's  declination  (given  by  the  almanac);  and  the  true 
distance  from  the  zenith  to  the  centre  of  the  sun  at  the  moment  when  its 
upper  edge  is  at  the  horizon  is  90°  50',  which  is  made  up  of  90°,+  16'  (the 
mean  semi-diameter  of  the  sun),  plus  34'  (the  mean  refraction  at  the  horizon). 
The  resulting  hour-angle  P,  corrected  for  the  equation  of  time,  gives  the  mean 
time  (local)  at  which  the  sun's  upper  limb  touches  the  horizon,  under  the 
average  circumstances  of  temperature  and  barometric  pressure.  If  it  is  very 
cold,  with  the  barometer  standing  high,  sunrise  will  be  accelerated,  or  sunset 
retarded,  by  a  considerable  fraction  of  a  minute.  If  the  sun  rises  or  sets 
over  the  sea-horizon,  and  the  observer's  eye  is  at  any  considerable  elevation 
above  the  sea-level,  the  dip  of  the  horizon  must  also  be  added  to  the  90°  50' 
before  making  the  computation. 

The  beginning  and  end  of  twilight  may  be  computed  in  the  same  way 
by  merely  substituting  108°  for  90°  60'. 


94  PROBLEMS   OF   PRACTICAL   ASTRONOMY. 

131.  To  Compute  the  Time  of  the  Rising  or  Setting  of  a  Star,  or 
of  the  Moon.  —  In  the  case  of  a  star  we  compute  its  hour-angle  at  the 
horizon  just  as  for  sunrise,  only  using  90°  34'  for  the  zenith-distance  instead 
of  90°  50 '.  The  hour-angle  added  to  the  star's  Right  Ascension  gives  the 
sidereal  time  of  its  setting  ;  by  subtracting  the  hour-angle  we  get  the  sidereal 
time  of  its  rising.  The  sidereal  times  are  then  converted  into  local  mean- 
time by  the  data  given  in  the  Almanac.  (Appendix,  Art.  1000.) 

The  rapid  motion  of  the  moon  complicates  the  problem  in  her  case,  and 
we  have  to  use  a  method  of  approximation.  We  begin  by  estimating  the 
Greenwich  time  of  moon-rise  as  nearly  as  we  can  without  actual  calculation. 
We  then  take  out  from  the  Almanac  the  moon's  Right  Ascension  and  Decli- 
nation for  that  moment  (the  Almanac  gives  the  data  for  every  hour).  With 
the  declination  and  the  latitude  of  the  place  we  compute  the  moon's  hour- 
angle,  taking  the  zenith-distance  as  89°  53',  since  the  horizontal  parallax  of 
the  moon  (57')  is  to  be  deducted  from  the  90°  50'  which  we  used  in  the 
case  of  the  sun.  The  hour-angle  thus  computed  is  then  subtracted  from  the 
moon's  Right  Ascension,  and  we  thus  get  an  approximate  sidereal  time  of 
moon-rise,  which  must  be  converted  into  mean  time.  If  the  time  originally 
assumed  by  estimation  does  not  differ  from  this  computed  result  by  more 
than  fifteen  minutes  or  so,  the  latter  may  be  taken  as  correct  within  a  frac- 
tion of  a  minute.  But  if  the  difference  is  greater,  we  must  have  recourse  to 
the  Almanac  again,  must  look  out  afresh  the  Right  Ascension  and  Declina- 
tion of  the  moon  corresponding  to  the  approximate  time,  as  computed,  and 
then  repeat  the  calculation  with  the  new  data.  A  third  computation  is 
never  necessary. 


EXERCISES  ON  CHAPTER  IV. 

(In  cases  where  corrections  for  refraction  are  given  they  are  to  be  taken 
from  Table  VIII,  Appendix,  taking  into  account  the  temperature  and  baro- 
metric pressure  if  given  among  the  data.) 

1.  Given  the  following  meridian  circle  observations  on  Beta  Ursae  Minoris 
at  its  upper  and  lower  culminations  respectively,  viz. : 

55°  48'  06.0",  Temp.  30°  F.,  Barometer  30.1  inches. 
24°  58'  56.4"       «       25°  F.          «  30.1      « 

The  nadir  reading  (Art.  67)  was  270°  01'  06.8"  in  both  cases.  Required 
the  latitude  of  the  place  and  the  declination  of  the  star. 

Lat.  40°  20'  57.8". 
'*   Dec.  74°  34' 40.1". 

2.  Given  the  meridian  altitude  of  the  sun's  lower  limb,  62°  24'  45",  the 
height  of  the  observer's  eye  above  the  sea-level  being  16  feet  (Art.  81). 

The  sun's  declination  was  +  20°  55'  10",  and  its  semidiameter,  15'  47". 
Its  parallax  at  the  observed  altitude  was  5",  and  the  mean  refraction  may 
be  used.  Required  the  latitude  of  the  ship.  Ans.  +  48°  19'  03". 


POSITION    OF    A   HEAVENLY   BODY. 


95 


3.  How  much  will  a  sidereal  clock  gain  on  a  mean  solar  clock  in  10 
hours  and  30  minutes  ? 

Ans.    1  min.  43.5  sec. 

4.  How  many  times  will  the  seconds  hand  of  a  sidereal  clock  overtake 
that  of  a  solar  clock  in  a  solar  day  if  they  start  together  ? 

Ans.   236  times. 

5.  At  what  intervals  do  the  coincidences  occur? 

Ans.   6  min.  5.242  sec. 

6.  In  determining  longitudes  by  telegraph  will  it  or  will  it  not  make  a 
difference  whether  sidereal  or  solar  clocks  are  used  by  the  observers  ? 

7.  A  ship   leaving.   San    Francisco   on   Tuesday,   October   12,   reaches 
Yokohama  after  a  passage  of  exactly  sixteen  days.     On  what  day  of  the 
month  and  of  the  week  does  she  arrive  ? 


—10- 


+10- 


+20- 


-10 


-30 


-1-30  -HO  0  —10 

FIG.  40.*  —  Path  of  the  Earth's  Pole  from  1890  to 


--30 


—30 
(Albrecht.) 


In  this  figure  the  scale  is  hundredths  of  a  second  of  arc,  each  of  which 
is  very  approximately  one  foot.      The  zero  at  the  bottom  indicates  the 


96  PROBLEMS    OF    PRACTICAL   ASTRONOMY. 

direction  of  Greenwich  from  the  "  mean  pole,'r  and  the  zero  at  the  left  hand 
(nearly)  that  of  Chicago  and  New  Orleans.  The  position  of  the  pole  is 
marked  for  each  third  month,  the  dotted  portions  of  the  curve  indicating- 
times  during  which  no  actual  observations  were  available. 

It  is  to  be  borne  in  mind  that  although  the  curve  is  based  on  observations 
at  more  than  a  dozen  different  stations,  yet  the  possible  error  of  the  plotted 
result  for  the  place  of  the  pole  at  a  given  moment  may  easily  be  four  or 
five  feet  in  error,  and  the  absolute  correctness  of  the  curve  must  not  be 
too  implicitly  accepted. 


THE  EARTH  AS  AN  ASTRONOMICAL  BODY.        97 


CHAPTER   V. 

THE   EARTH   AS   AN   ASTRONOMICAL   BODY. 

APPROXIMATE    DIMENSIONS.  PROOFS     OF     ITS     ROTATION.  - 

ACCURATE     DETERMINATION     OF     ITS     FORM     AND     SIZE    BY 
GEODETIC    OPERATIONS    AND    PENDULUM    OBSERVATIONS.  - 
ASTRONOMICAL,    GEODETIC    AND    GEOCENTRIC    LATITUDE.  - 
DETERMINATION    OF   THE   EARTH'S   MASS   AND   DENSITY. 

132,  HAVING  discussed  the  methods  of  making  astronomical  ob- 
servations, we  are  now  prepared  to  consider  the  earth  in  its  astro- 
nomical relations ;  i.e.,  those  facts  relating  to  the  earth  which  are 
ascertained  by  astronomical  methods,  and  are  similar  to  the  facts 
which  we  shall  have  to  consider  in  the  case  of  the  other  planets. 
The  facts  are  broadly  these  :  — 

1.  The  earth  is  a  great  ball,  about  7918  miles  in  diameter. 

2.  It  rotates  on  its  axis  once  in  twenty-four  sidereal  hours. 

3.  It  is  flattened  at  the  poles,  the  polar  diameter  being  nearly 
twenty-seven  miles,  or  one  two  hundred  and  ninety-fifth  part  less  than 
the  equatorial. 

4.  It  has  a  mean  density  between  5.5  and  5.6  as  great  as  that  of 
water,  and  a  mass  represented  in  tons  by  six  with  twenty-one  ciphers 
after  it  (or  six  sextillions  of  tons,  according  to  the  French  numeration). 

5.  It  is  flying  through  space  in  its  orbital  motion  around  the  sun, 
with  a  velocity  of  about  nineteen  miles  a  second  ;  i.e.,  about  seventy- 
five  times  as  swiftly  as  any  cannon-ball. 


133.  The  Earth's  Approximate  Form  and  Size.  —  It  is  not  neces- 
sary to  dwell  upon  the  ordinary  proofs  of  its  globularity.  We  merely 
mention  them.  1.  It  can  be  circumnavigated.  2.  The  appearance  of 
vessels  coming  in  from  sea  indicates  that  the  surface  is  everywhere 
convex.  3.  The  fact  that  the  sea-horizon,  as  seen  from  an  emi- 
nence, is  everywhere  depresse4  to  the  same  extent  below  the  level 
line,  shows  that  the  surface  is  approximately  spherical.  4.  The  fact 
that  as  one  goes  from  the  equator  toward  the  north,  the  elevation  of 


98  THE   EARTH   AS   AN   ASTRONOMICAL   BODY. 

the  pole  increases  proportionally  to  the  distance  from  the  equator 
proves  the  same  thing.  5.  The  shadow  of  the  earth,  as  seen  upon 
the  moon  at  the  time  of  a  lunar  eclipse,  is  that  which  only  a  sphere 
could  cast. 

We  may  add  as  to  the  smoothness  and  globularity  of  the  earth, 
that  if  the  earth  be  represented  by  an  18-inch  globe,  the  difference 
between  the  polar  and  equatorial  diameter  would  only  be  about  one- 
sixteenth  of  an  inch,  the  highest  mountains  upon  the  earth's  surface 
would  be  represented  by  about  one-eightieth  of  an  inch,  and  the 
average  elevation  of  the  continents  would  be  hardly  greater  than 
that  of  a  film  of  varnish.  The  earth  is  really  relatively  smoother 
and  rounder  than  most  of  the  balls  in  a  bowling-alley. 

134.  The  best  method  of  ascertaining  the  size  of  the  earth  —  in 
fact  the  only  one  of  real  value  —  is  by  measuring  arcs  of  the  merid- 
ian in  order  to  ascertain  the  number  of  miles  or  kilometres  in  one 
degree,  from  which  we  immediately  get  the  circumference  of  the 
earth.     This  measure  involves  two  distinct  operations.     One  —  the 
measure  of  the  number  of  miles  —  is  purely  geodetic  ;  the  other  — 
the  determination  of  the  number  of  degrees,  minutes,  and  seconds 
between  the  two  stations  —  is  purely  astronomical. 

We  have  to  find  by  astronomical  observation  the  angle  between 
two  radii  drawn  from  the  centre  of  the  earth  to  the  two  stations 
(regarding  the  earth  as  spherical)  ;  or,  what  is  the  same  thing,  the 
angular  distance  in  the  sky  between  their  respective  zeniths.  The  two 
stations  being  on  the  same  meridian,  all  that  is  necessary  is  to  meas- 
ure their  latitudes  by  any  of  the  methods  which  have  been  given  in 
Chapter  IV.  and  take  the  difference.  This  will  be  the  angle  wanted. 
If,  for  instance,  the  distance  between  the  two  stations  was  found  by 
measurement  to  be  120  miles,  and  the  difference  of  latitude  was 
found  by  astronomical  observations  to  be  1°  44 '.2,  we  should  get 
69.27  miles  for  one  degree.  Three  hundred  and  sixty  times  this 
would  be  the  circumference  of  the  earth,  a  little  less  than  25,000 
miles,  and  the  diameter  would  be  found  by  dividing  this  by  TT,  which 
would  give  7920  miles. 

135.  Eratosthenes  of  Alexandria  seems  to  have  understood  the  matter 
as  early  as  250  B.C.    His  two  stations  were  Alexandria  and  Syene  in  Upper 
Egypt.     At  Syene  he  observed  that  at  noon  of  the  longest  day  in  summer 
there  was  no  shadow  at  the  bottom  of  a  well,  the  sun  being  then  vertically 
overhead.     On  the  other  hand,  the  gnomon  at  Alexandria,  on  the  same  day, 
by  the  length  of  the  shadow,  gave  him  ^  of  a  circumference,  or  7°  12'  as 


SIZE    OF    THE    EARTH    REGARDED    AS    A    SPHERE. 


99 


the  distance  of  the  sun  from  the  zenith  at  that  place,  which,  therefore,  is 
the  difference  of  latitude  between  Alexandria  and  Syene.  , 

The  weak  place  in  his  work  was  in  the  measurement  of  the  distance  be- 
tween the  two  places.  He  states  it  as  5000  stadia,  thus  making  the  circum- 
ference of  the  earth  250,000  stadia  ;  but  we  do  not  know  the  length  of  his 
stadium,  nor  does  he  give  any  account  of  the  means  by  which  he  measured 
the  distance,  if  he  measured  it  at  all.  There  seem  to  have  been  as  many 
different  stadia  among  the  ancient  nations  as  there  were  kinds  of  «  feet "  in 
Europe  at  the  beginning  of  this  century. 

The  first  really  valuable  measure  of  the  arc  of  a  meridian  was  that  made 
by  Picard  in  Northern  France  in  1671  —  the  measure  which  served  Newton 
so  well  in  his  verification  of  the  idea  of  gravitation. 

136.  An  approximate  measure  of  the  diameter  is  easily  obtained. 
Erect  upon  a  level  plain 

three  rods  in  line,  a  mile  A  -B ^ 

apart,  and  cut  off  their 
tops  at  the  same  level, 
carefully  determined 
with  a  surveyor's  level- 
ling instrument.  It  will 
then  be  found  that  the 
line  AC,  Fig.  44,  join- 
ing the  extremities  of  the  two  terminal  rods,  when  corrected  for  refraction, 
passes  about  eight  inches  below  B,  the  top  of  the  middle  rod. 

Suppose  the  circle  ABC  completed,  and  that  E  is  the  point  on  the  cir- 
cumference opposite  B,  so  that  BE  equals  the  diameter  of  the  earth  (=2/2). 

By  geometry,     BD  :  BA  =  BA  :  BE, 

BA* 


FIG.  44.  —  Curvature  of  the  Earth's  Surface. 


whence 


BD 


,  or  K  — 


2BD 


Now  BA  is  one  mile,  and  BD  =  §  of  a  foot,  or 
I2 


Hence  2  R  — 


7920 


of  a  mile. 
,  or  7920  miles  :  a  very  fair  approximation. 


On  account  of  refraction,  however,  the  result  cannot  be  made  exact  by 
any  care  in  observation.  The  observed  value  of  BD  (uncorrected)  ranges 
from  4.5  inches  to  6.5,  according  to  the  state  of  the  weather. 

II. 

137.  The  Rotation  of  the  Earth.  —  At  the  time  of  Copernicus  the 
only  argument  in  favor  of  the  earth's  rotation  l  was  that  the  hypoth- 

1  The  word  rotate  denotes  a  spinning  motion  like  that  of  a  wheel  on  its  axis. 
The  word  revolve  is  more  general  in  its  application,  and  may  be  applied  either  to 


100  THE   EARTH   AS   AN   ASTRONOMICAL   BODY. 

esis  was  more  probable  than  that  the  heavens  themselves  revolved. 
All  phenomena  then  known  would  be  sensibly  the  same  on  either 
supposition.  A  little  later,  analogy  could,  be  adduced,  for  when  the 
telescope  was  invented,  we  could  see  that  the  sun,  moon,  and  several 
of  the  planets  are  rotating  globes. 

At  present  we  are  able  to  adduce  experimental  proofs  which  abso- 
lutely demonstrate  the  earth's  rotation,  and  some  of  them  even  make 
it  visible. 

138.     1.   The  Eastward  Deviation  of  Bodies  fatting  from  a  Great 
Height. — The  idea  that  such   a  deviation   ought  to  occur  was  first 
suggested  by  Newton.     Evidently,  since  the  top  of  a  tower,  situated 
anywhere  but  at  the  pole  of  the  earth,  describes  every 
day  a  larger  circle  than  its  base,  it  must  move  faster. 
A  body  which  is  dropped  from  the  top,  retaining  its  ex- 
cess of  eastward  motion  as  it  descends,  must  therefore 
strike  to  the  east  of  the  point  which  is  vertically  under 
its  starting-point,  provided  it  is  not  deflected  in  its  fall 
by  the  resistance  of  the  air  or  by  air-currents.     Fig.  45 
illustrates  the  principle.     A  body  starting  from  A,  the 
top   of    the   tower,   reaches   the  earth   at   D  (BD   be- 
ing  equal  very  approximately  to   AA'),  while   during 
its  fall    the  bottom  of  the  tower  has  only  moved  from 
FIG.  45.        B  to  B'.     The  experiments  are  delicate,  since  the  devi- 
Eastward  Devia-  ation  is    very  small,  and  it  is  not  easy  to    avoid   the 
uon  of  a  Failing  effectof  air_currents.     jt  js   aiso  extremely  difficult  to 

get  balls  so  perfectly  spherical  that  they  will  not  sheer 
off  to  one  side  or  the  other  in  falling. 

The  best  experiments  of  this  kind  so  far  have  been  those  of  Benzenberg, 
performed  at  Hamburg  in  1802,  and  those  of  Reich,  performed  in  1831,  in 
an  abandoned  mine  shaft  near  Freiberg,  in  Saxony.  The  latter  obtained  a 
free  fall  of  520  feet,  and  from  the  mean  of  106  trials,  the  eastern  deviation 
observed  was  1.12  inches,  while  theory  would  make  it  1.08.  The  experiment 
also  gave  a  southern  deviation  of  0.17  of  an  inch,  unexplained  by  theory. 
It  seems  to  indicate  the  probable  error  of  observation.  The  balls  in  falling 
sometimes  deviated  two  or  three  inches  one  side  or  the  other  from  the 
average. 

describe  such  a  spinning  motion,  or  (and  this  is  the  more  usual  use  in  astronomy) 
to  describe  the  motion  of  one  body  around  another,  as  that  of  the  earth  around 
the  sun. 


PROOFS  OF  THE  EARTH'S  ROTATION. 


101 


The  formula  given   by  Worms  in  his  treatise  on  "  The  Earth  and  its 
Mechanism,"  is 


where  x  is  the  deviation,  t  is  the  number  of  seconds  occupied  in  falling,  T 
the  number  of  seconds  in  a  sidereal  day,  H  the  height  fallen  through,  and 
A  the  difference  between  H  and  the  height  through  which  a  body  would  fall 
In  t  seconds  if  there  were  no  resistance  (so  that  A  =  §  gt2—H).  Finally,  0  is 
the  latitude  of  the  place  of  observation.  In  latitude  45°  a  fall  of  576  feet 
should  give,  neglecting  the  resistance  of  the  air,  a  deviation  of  1.47  inches. 
The  resistance  would  increase  it  a  little. 

It  will  be  noted  that  at  the  pole,  where  the  cosine  of  the  latitude  equala 
•zero,  the  experiment  fails.    The  largest  deviation  is  obtained  at  the  equator. 

f  139.  2.  FoucauWs  Pendulum  Experiment.  —  In  1851  Foucault, 
that  most  ingenious  of  French 
physicists,  devised  and  first  exe- 
cuted an  experiment  which  actually 
shows  the  earth's  rotation  to  the 
eye.  From  the  dome  of  the  Pan- 
theon in  Paris  he  suspended  a  heavy 
iron  ball  about  a  foot  in  diameter 
by  a  wire  more  than  200  feet  long 
(Fig.  46).  A  circular  rail  some 
twelve  feet  across,  with  a  little 
ridge  of  sand  built  upon  it,  was 
placed  under  the  pendulum  in  such 
a  way  that  a  pin  attached  to  the 
swinging  ball  would  just  scrape 
the  sand  and  leave  a  mark  at  each 
vibration.  The  ball  was  drawn 
aside  by  a  cotton  cord  and  allowed 
to  come  absolutely  to  rest;  then 
the  cord  was  burned,  and  the  pen- 

dulum set  to  Swinging  in  a  true  FIG.  46.—  Foucault's  Pendulum  Experiment 
plane  ;  but  this  plane  seemed  to 

deviate  slowly  towards  the  right,  cutting  the  sand  in  a  new  place  at 
each  swing  and  shifting  at  a  rate  which  would  carry  io  completely 
around  in  about  thirty-two  hours  if  the  pendulum  did  not  first  come 
to  rest.  In  fact,  the  floor  of  the  Pantheon  was  seen  turning  under 
the  plane  of  the  pendulum's  vibration.  The  experiment  created 


102 


THE   EARTH   AS  AN   ASTRONOMICAL   BODY. 


great  enthusiasm  at  the  time,  and  has  since  been  very  frequently 
performed,  and  always  with  substantially  the  same  results. 

140.  The  approximate  theory  of  the  experiment  is  very  simple. 
Such  a  pendulum,  consisting  of  a  round  ball  hung  by  a  round  wire  or 
else  suspended  on  a  point,  so  as  to  be  equally  free  to  swing  in  any  plane 
(unlike  the  common  clock  pendulum  in  this  freedom) ,  being  set  up 
at  the  pole  of  the  earth,  would  appear  to  shift  around  in  twenty-four 
hours.     Really,  the  plane  of  vibration  remains   invariable  and  the 
earth  turns  under  it,  the  plane  of  vibration  in  this  case  being  un- 
affected  by  the   motion   of  the   earth.     This  can  be  easily  shown 
by  setting  up  a  similar  apparatus,  consisting  of  a  ball  hung  by  a 
thread,  upon  a  table,  and  then  turning  the  table  around  with  as  little 
jar  as  possible.     The  plane  of  the  swing  will  remain  unchanged  by 
the  motion  of  the  table. 

It  is  easy  to  see,  further,  that  at  the  equator  there  would  be  no  such 
tendency  to  shift.  In  any  other  latitude  the  effect  will  be  intermedi- 
ate, and  the  time  required  for  the  pendulum  to  complete  the  revolu- 
tion of  its  plane  will  be  twenty-four  hours 
divided  by  the  sine  of  the  latitude.  The  north- 
ern edge  of  the  floor  of  a  room  (in  the  northern 
hemisphere)  is  nearer  the  axis  of  the  earth 
than  its  southern  edge,  and  therefore  is  car- 
ried more  slowly  eastward  by  the  earth's  rota- 
tion. Hence  it  must  skew  around  continually, 
like  a  postage  stamp  gummed  upon  a  whirling 
globe  anywhere  except  at  the  globe's  equator. 
The  southern  extremity  of  every  north  and 
south  line  on  the  floor  continually  works  to- 
ward the  east  faster  than  the  northern  ex- 
tremity, causing  the  line  itself  to  shift  its  direc- 
tion accordingly,  compared  with  the  direction 
it  had  a  few  minutes  before.  A  free  pendu- 
lum, set  at  first  to  swing  along  such  a  line,  must  therefore  apparently 
deviate  continually  at  the  same  rate  in  the  opposite  direction.  In 
the  northern  hemisphere  its  plane  moves  dextrorsum;  i.e.,  with  the 
hands  of  a  watch :  in  the  southern,  its  motion  is  sinistrorsum. 

141.  Suppose  a  parallel  of  latitude  drawn  through  the  place  in  question, 
and  a  series  of  tangent  lines  drawn  toward  the  north  at  points  an  inch  or  so 
apart  on  this  parallel.     All  these  tangents  would  meet  at  some  point,  F,  Fig. 
47,  which  is  on  the  earth's  axis  produced  ;  and  taken  together  these  tangents 


PIG.  47. 

Explanation  of  the  Foucault 
Pendulum  Experiment. 


PEOOFS   OF   THE   EARTH'S   ROTATION.  103 

would  form  a  cone  with  its  point  at  V.     Now  if  we  suppose  this  cone  cut 

down  upon  one  side  and  opened  up  (technically,  "  developed  "),  it  would  give 

us  a  sector  of  a  circle,  as  in  Fig.  48,  and  the 

angle  V,  reckoned  around  from  A  to  A'  through 

B,  is  the  sum  total  of  all  the  angles  between 

all  the  adjacent  meridian  tangents  touching  the 

earth    on    that    parallel   (—  a  reentrant   angle, 

greater  than   180°,  in  the  figure).     Now,  fast, 

the  circumference  of  the  parallel  (Fig.  47),  or 

the  arc  ABA',  which  measures  the  angle  V  in 

Fig.  48,  equals  2ir  X  AD  ;  and,  since  the  angle 

DAC  (Fig.  47)  equals  the   latitude,  AD  =  R 

cos  <£  (<#>  being  the  latitude).     Hence  ABA'  =  ^"^-—  - 

2  TT  R  X  cos  <j>.     Second,  the  radius  of  the  sector      FlG  43.—  Developed  Cone. 

in  Fig.  48  is  the  same  as  A  V,  the  side  of  the 

cone  in  Fig.  47  ;  and  since  in  Fig.  47  the  angle  A  VD  =  <£,  we  have  A  V  = 

R  cot  <j>,  and  the  circumference,  ABA'm  =2  TT  R  cot  </>. 


V        ABA'         2 
Hence,>a%,  —  =  --       =  =  sin  *  and  V=  360°  S1n 


i.e.,  the  total  angle  described  by  the  plane  of  the  pendulum  in  a  day  =  360° 
X  sin  of  the  latitude. 

At  the  pole  the  cone  produced  by  the  tangent  lines  becomes  a  little 
"  button,"  a  complete  circle.  At  the  equator  it  becomes  a  cylinder,  and  the 
angle  is  zero. 

It  is  worth  noting  that  the  azimuthal  motion  of  any  star  at  the  horizon 
in  a  minute  of  time  is  15'  X  sin  <f>,  —  the  same  at  all  parts  of  the  horizon. 
See  Appendix,  Art.  1001. 

In  order  to  make  the  experiment  successfully,  many  precautions  must  be 
taken.  It  is  specially  important  that  the  pendulum  should  vibrate  in  a  true 
plane,  without  any  lateral  motion.  To  secure  this  end,  it  must  be  carefully 
guarded  against  all  jarring  motion  and  air-currents.  To  diminish  the  effect 
of  all  such  disturbances,  which  will  always  occur  to  a  certain  extent,  the 
pendulum  should  be  very  heavy  and  very  long,  and  of  course  the  suspended 
ball  must  be  truly  round  and  smooth.  Ordinary  clock-work  cannot  be  used 
to  keep  the  pendulum  in  vibration,  since  it  must  be  free  to  swing  in  every 
plane.  Usually,  the  apparatus  once  started  is  left  to  itself  until  the  vibra- 
tions cease  of  their  own  accord  ;  but  Foucault  contrived  a  most  ingenious 
electrical  apparatus,  which  we  have  not  space  to  describe,  by  means  of  which 
the  vibration  could  be  kept  up  for  days  at  a  time  without  receiving  any 
hurtful  disturbance  whatever. 

It  will  be  noticed  that  this  experiment  is  most  effective  precisely  where 
the  experiment  of  falling  bodies  fails.  This  is  best  near  the  pole,  the  other 
at  the  equator. 

142.  3.  By  the  Gyroscope,  an  experiment  also  due  to  Foucault, 
and  proposed  and  executed  soon  after  the  pendulum  experiment. 


104 


THE  EARTH   AS   AN   ASTRONOMICAL  BODY. 


The  instrument  shown  in  Fig.  49  consists  of  a  wheel  so  mounted  in 
gimbals  that  it  is  free  to  turn  in  every  direction,  and  so  delicately 
balanced  that  it  will  stay  in  any  position  if  undisturbed.  If  the 
wheel  be  set  to  rotating  rapidly,  it  will  maintain  the  direction  of  its 
axis  invariable,  unless  acted  upon  by  extraneous  force.  If,  then, 
we  set  the  axis  horizontal  and  arrange  a  microscope  to  watch  a 

mark  upon  one  of  the  gim- 
bals, it  will  appear  slowly  to 
shift  its  position  as  the  earth 
revolves,  in  the  same  way  as 
the  plane  of  the  pendulum 
behaves. 

143.  4.  There  are  many  other 
phenomena  which  depend  upon 
and  really  demonstrate  the  earth's 
rotation.  We  merely  mention 
them :  — 

a.  The  Deviation  of  Projectiles. 
In  the  northern  hemisphere  a 
projectile  always  deviates  towards 
the  right ;  in  the  southern  hemi- 
sphere toward  the  left. 
5.  The  Trade  Winds, 
c.  The  Vorticose  Revolution  of 
the  Wind  in  Cyclones.  In  the 
northern  hemisphere  the  wind  in 
a  cyclone  moves  spirally  towards 
the  centre  of  the  storm,  whirling 
counter  clock-wise,  while  in  the 

southern  the  spiral  motion  is  with  the  hands  of  a  watch.  The  motion  is 
explained  in  either  case  by  the  fact  that  currents  of  air,  setting  out  for  the 
centre  of  disturbance  where  the  cyclone  is  formed,  deviate  like  projectiles, 
to  the  right  in  the  northern  hemisphere,  and  towards  the  left  in  the  southern 
hemisphere,  so  that  they  do  not  meet  squarely  in  the  centre  of  disturbance. 
d.  The  Ordinary  Law  of  Wind-change ;  that  is,  in  the  northern  hemisphere 
the  north  wind,  under  ordinary  circumstances,  changes  to  a  northeast,  a 
northeast  wind  to  an  east,  east  to  southeast,  etc.  When  the  wind  changes 
in  the  opposite  direction,  it  is  said  to  "back''  around.  In  the  southern 
hemisphere  it  of  course  usually  backs  around,  much  to  the  disconcertment 
of  the  early  Australian  settlers. 

It  might  seem  at  first  that  the  rotation  of  the  earth,  which  occupies 
twenty-four  hours,  is  not  a  very  rapid  motion.     A  point  on  the  equa- 


FIG.  49.  — Foucault's  Gyroscope. 


SPHEROIDAL    FORM    OF   THE    EARTH.  105 

tor,  however,  has  to  move  nearly  one  thousand  miles  an  hour,  which 
is  about  fifteen  hundred  feet  per  second,  and  very  nearly  the  speed 
of  a  cannon-ball. 

144.  Invariability  of  the  Earth's  Rotation.  — It  is  a  question  of 
great  importance  whether  the  day  changes  its  length.     Theoretically 
it  must  almost  necessarily  do  so.      The  friction  of  the  tides  and  the 
deposits  of  meteoric  matter  upon  the  earth  both  tend  to  lengthen  it ; 
while  on  the  other  hand,  the  earth's  loss  of  heat  by  radiation  and 
consequent  shrinkage   must  tend  to  shorten  it.      Then  geological 
changes,  the  elevation  and  subsidence  of  continents,  and  the  trans- 
portation of  matter  by  rivers,  act,  some  one  way,  some  the  other.     At 
present  it  can  only  be  said  that  the  change,  if  any  has  occurred  since 
astronomy  became  accurate,  has  been  too  small  to  be  detected.    The 
day  is  certainly  not  longer  or  shorter  by  T£<y  of  a  second  than  in  the 
days  of  Ptolemy,  and  probably  has  not  changed  by  T ^  of  a  second. 
The  criterion  is  found  in  comparing  the  times  at  which  celestial 
phenomena,  such  as  eclipses,  transits  of  Mercury,  etc.,  occur.     For 
changes  in  the  position  of  the  axis,  see  Art.  108. 

III. 

145.  The  Earth's  Form,  more   accurately  stated,  is  that  of  a 
spheroid  of  revolution,  having  an  equatorial  radius  of  6,377,377  me- 
tres, and  a  polar  radius  of  6,355,270  metres,  according  to  Listing 
(1873)  ;   or  of  6,378,206.4  and  6,356,583.8  respectively,  according  to 
Clarke.1     It  must  be  understood,  also,  that  this  statement  is  only  a 
second  approximation  (the  first  being  that  the  earth  is  a  globe). 
Owing  to  mountains  and  valleys,  etc.,  the  earth's  surface  does  not 
strictly  correspond  to  that  of  any  geometrical  solid  whatever. 

The  flattening  at  the  poles  is  the  necessary  consequence  of  the 
earth's  rotation,  and  might  have  been  cited  in  the  preceding  section 
as  proving  it. 

146.  There  are  three  ways  of  determining  the  form  of  the  earth  : 
one,  by  measurement  of  distances  upon  its  surface  in  connection  with 
the  latitudes  and  longitudes  of  the  points  of  observation.     This  gives 
not  only  the  form,  but  the  dimensions.     The  second  method  is  by  the 
observation  of  the  varying  force  of  gravity  at  various  points,  —  ob- 
servations which  are  made  by  means  of  a  pendulum  apparatus  of  some 

1  This  is  Clarke's  spheroid  of  1866,  and  is  adopted  by  the  United  States  Coast 
and  Geodetic  Survey.  See  Appendix,  page  601,  for  his  spheroid  of  1878. 


106 


THE  EARTH   AS   AN   ASTRONOMICAL   BODY. 


kind,  and  determine  only  the  form,  but  not  the  size  of  the  earth. 
The  third  method  is  by  means  of  certain  purely  astronomical  phe- 
nomena, known  as  " precession"  and  "nutation"  (to  be  treated  of 
hereafter),  and  by  certain  irregularities  in  the  motion  of  the  moon. 
Observations  of  the  occultations  of  stars  at  widely  distant  stations  can 
also  be  utilized  for  the  same  purpose.  All  the  methods  of  this  third 
class,  like  the  pendulum  method,  give  only  the  form  of  the  earth. 

147.  1.  Measurements  of  Arcs  of  Meridian  in  Different  Latitudes. 
—  To  determine  the  size  of  the  earth  regarded  as  a  sphere,  a  single 
arc  of  meridian  in  any  latitude  is  sufficient.  Assuming,  however, 
that  the  earth  is  not  a  sphere,  but  a  spheroid  with  elliptical  meridians, 
we  must  measure  at  least  two  such  arcs,  one  of  which  should  be  near 
the  equator,  the  other  near  the  pole. 

The  astronomical  work  consists  simply  in  finding  with  the  greatest 
possible  accuracy  the  difference  of  latitude  between  the  terminal  sta- 
tions of  the  meridian  arc.  The  geodetic  work  consists  in  measuring 
their  distance  from  each  other  in  miles,  feet,  or  metres,  and  it  is  this 
part  of  the  work  which  consumes  the  most  time  and  labor.  The 
process  is  generally  that  known  as  triangulation. 

Two  stations  are  selected  for  the  extremities  of  a 
base  line  six  or  seven  miles  long,  and  the  ground 
between  them  is  levelled  as  if  for  a  railroad.  The 
distance  between  these  stations  (A  and  B  in  Fig.  50) 
is  then  carefully  measured  by  an  apparatus  especially 
designed  for  the  purpose  and  with  an  error  not  to 
exceed  half  an  inch  or  so  in  the  whole  distance.  A 
third  station,  1,  is  then  chosen,  so  situated  that  it  will 
be  visible  from  both  A  and  J3,  and  all  the  angles  of 
the  triangle  AB  1  are  measured  with  great  care  by  a 
theodolite.  A  fourth  station,  2,  is  then  selected,  such 
that  it  will  be  visible  from  A  and  1  (and  if  possible 
from  B  also),  and  the  angles  of  the  triangle  .412  are 
measured  in  the  same  way.  In  this  manner  the  whole 
ground  between  the  two  terminal  stations  is  covered 

with  a  network  of  triangulation,  the  two  terminal  stations  themselves  being 
made  two  of  the  triangulation  points.  Knowing  one  distance  and  all  the 
angles  in  this  system,  it  is  possible  to  compute  with  great  accuracy  the  exact 
length  of  the  line  1  5  and  its  direction. 


FIG.  50.  — A  Triangulation. 


The  sides  of  the  triangles  are  usually  from  twenty-five  to  thirty 
miles  in  length,  though  in  a  mountainous  country  not  infrequently 
much  longer  ones  are  available.  Generally  speaking,  the  fewer  the 


DIMENSIONS  OF  THE  EARTH  FROM  GEODETIC  MEASURES.    107 

stations  necessary  to  connect  the  extremities  of  the  arc,  and  the 
longer  the  lines,  the  greater  will  be  the  ultimate  accuracy.  In  this 
way  it  is  possible  to  measure  distances  of  200  or  300  miles  with  a 
probable  error  not  exceeding  two  or  three  feet. 

Many  arcs  of  meridians  have  been  measured  in  this  way,  —  not  less  than 
twenty  or  thirty  in  different  parts  of  the  earth,  the  most  extensive  being  the 
so-called  Anglo-French  arc,  extending  more  than  twelve  degrees  in  length  ; 
the  Indian  arc,  nearly  eighteen  degrees  long ;  and  the  great  Russo-Scandi- 
navian  arc,  more  than  twenty-five  degrees  in  length,  and  reaching  from 
Hammerfest  to  the  mouth  of  the  Danube.  One  short  arc  has  been  measured 
in  South  America  and  one  in  South  Africa. 

In  a  general  way,  it  appears  that  the  higher  the  latitude  the 
longer  the  arc.  Thus,  near  the  equator  the  length  of  a  degree  has 
been  found  to  be  362,800  feet  in  round  numbers,  while  in  northern 
Sweden,  in  latitude  66°,  it  is  365,800  feet;  in  other  words,  the 
earth's  surface  is  flatter  near  the  poles.  It  is  necessary  to  travel 
3000  feet  further  in  Sweden  than  in  India  to  increase  the  latitude 
one  degree,  as  measured  by  the  elevation  of  the  celestial  pole. 

The  following  little  table  gives  the  length  of  a  degree  of  the  meridian  at 
different  latitudes  :  — 

At  the  equator  one  degree  =  68.704  miles. 
At  latitude  20°   «        «       =  68.786      " 

"        "        40°   "        "       =  68.993      " 

"        "        60°   "        "       =  69.230      " 

«        "        80°   "        "       =  69.386      " 

"        «        90°   "        "       =  69.407      " 

The  difference  between  the  equatorial  and  polar  degree  of  latitude  is 
more  than  seven-tenths  of  a  mile,  or  over  3500  feet,  while  the  probable 
error  of  measurement  cannot  exceed  more  than  a  foot  or  two  to  the  degree. 

It  will  be  understood,  of  course,  that  the  length  of  a  degree  at  the  pole 
is  obtained  by  extrapolation  from  the  measures  made  in  lower  latitudes. 

148.  The  deduction  of  the  exact  form  of  the  earth  from  such 
measurements  is  an  abstruse  problem.  Owing  to  errors  of  observa- 
tion and  local  deviations  in  the  direction  of  gravity,  the  different  arcs 
do  not  give  strictly  accordant  results,  and  the  best  that  can  be  done 
is  to  find  the  result  which  most  nearly  satisfies  all  the  observations. 

If  we  assume  that  the  form  is  that  of  an  exact  spheroid  of  revolution, 
with  all  the  meridians  true  ellipses  and  all  exactly  alike,  the  problem 
is  simplified  somewhat,  though  still  too  complicated  for  discussion 


108  THE   EARTH   AS   AN   ASTRONOMICAL   BODY. 

here.  Theory  indicates  that  the  form  of  a  revolving  mass,  fluid 
enough  to  yield  to  the  forces  acting  in  such  a  case,  might,  and  prob- 
ably would,  be  such  a  spheroid;  but  other  forms  are  also  theoretically 
possible,  and  some  of  the  measurements  rather  indicate  that  the 
equator  of  the  earth  is  not  a  true  circle,  but  an  oval  flattened  by 
nearly  half  a  mile.  On  the  whole,  however,  astronomers  are  dis- 
posed to  take  the  ground  that  since  no  regular  geometrical  solid 
whatsoever  can  absolutely  represent  the  form  of  the  earth,  we  may 
as  well  assume  a  regular  spheroid  for  the  standard  surface,  and  con- 
sider all  variations  from  it  as  local  phenomena,  like  hills  and  valleys. 

149.  Each  measurement  of  a  degree  of  latitude  gives  the  "radius  of 
curvature"  as  it  is  called,  of  the  meridian  at  the  degree  measured.     The 
length  of  a  degree  from  44°  30'  to  45°  30',  multiplied  by  57.29  (the  number 
of  degrees  in  a  radian),  gives  the  radius  of  the  "osculatory  circle"  which 
would  just  fit  the  curve  of  the  meridian  at  that  point.     Having  a  table  giv- 
ing the  actual  length  of  each  degree  of  latitude,  we  could  construct  the 
earth's  meridian  graphically  as  follows  :  — 

Draw  the  line  AX,  Fig.  51.  On  it  lay  off  A  a,  equal  to  the  radius  of  curva- 
ture of  the  first  measured  degree  (that  is,  57.3  times  the  length  of  the  degree), 
and  with  a  as  centre,  describe  an  arc  AB,  making  the  angle  AaB  just  one 

degree.  Next  produce  the  line  Ba  to  6, 
making  Bb  the  radius  of  curvature  of 
the  second  degree,  and  draw  this  second 
degree-arc  ;  and  so  proceed  until  the 
whole  ninety  have  been  drawn.  This 
will  give  one-quarter  of  the  meridian, 
and  of  course  the  three  other  quarters 
are  all  just  like  it.  a,  b,  c,  etc.,  are  called 
the  "centres  of  curvature"  of  the  differ- 
ent degrees. 

If  we  assume  the  curve  to  be  an  el- 
lipse, then  the  equatorial  semidiameter, 
AO,  and  the  polar,  PO,  are  given  respec- 

FIG<  51<  tively  by  the  two  formulas,  A  0  =  %fqp* 

Radii  of  Curvature  of  the  Meridian.  _   __.         8/T"  11-        ,*  T- 

and  PO  =  v^p,  q  and  p  being  the  radii 

of  curvature  (Aa  and  Pe  in  the  figure)  at  the  equator  and  pole. 

150.  The  "ellipticity"  or  "oblateness"  of  an  ellipse  is  the  frac- 
tion found  by  dividing  the  difference  of  the  polar  and  equatorial 
diameters  by  the  equatorial,  and  is  expressed  by  the  equation 


PENDULUM  EXPERIMENTS.  109 

In  the  case  of  the  earth  this  is  ^ig,  according  to  Clarke's  spheroid, 
of  1866.  Until  within  the  last  few  years  Bessel's  smaller  value, 
viz.,  2^9,  was  generally  adopted.  Listing's  larger  value,  gisj  is 
now  preferred  by  some. 

The  ellipticity  of  an  ellipse  must  not  be  confounded  with  its  eccen- 
tricity.    The  latter  is  _ 


and  is  always  a  much  larger  numerical  quantity  than  the  ellipticity. 
In  the  case  of  the  earth's  meridian,  it  is  T£  T  as  against  ^^.  Its 
symbol  is  usually  e. 

151.  Arcs  of  longitude  are  also  available  for  determining  the  earth's 
form  and  size.     On  a  spherical  earth  a  degree  of  longitude  measured  along 
any  parallel  of  latitude  would  be  equal  to  one  degree  of  the  equator  multi- 
plied by  the  cosine  of  the  latitude.    On  an  oblate  or  orange-shaped  spheroid 
(the  surface  of  which  lies  wholly  within  the  sphere  having  the  same  equator) 
the  degrees  of  longitude  are  evidently  everywhere  shorter  than  on  the  sphere, 
the  difference  being  greatest  at  a  latitude  of  45°. 

In  fact,  arcs  in  any  direction  between  stations  of  which  both  the  latitude  and 
longitude  are  known  can  be  utilized  for  the  purpose  ;  and  thus  the  extensive 
surveys  that  have  been  made  in  different  countries  have  given  us  a  pretty 
accurate  knowledge  of  the  earth's  dimensions.  It  is  very  desirable  that  in 
some  way  the  chain  of  actual  measurements  should  be  extended  from  the 
eastern  continent  to  the  western,  but  the  immense  difficulties  of  so  doing 
are  obvious. 

At  present  the  distance  from  a  point  on  the  earth's  surface  (say  the  ob- 
servatory at  Washington)  to  any  other  point  in  the  opposite  hemisphere 
(say  the  observatory  at  the  Cape  of  Good  Hope)  is  uncertain  to  perhaps  the 
extent  of  a  quarter  of  a  mile. 

152.  2.  Pendulum  Experiments.  —  Since  (Physics,  p.  75), 


=  v  \  " 


we  can  therefore  measure  the  variations  of  the  force  of  gravity,  g, 
at  different  parts  of  the  earth,  either  by  taking  a  pendulum  of  in- 
variable length  and  determining  t,  the  time  of  its  vibration  ;  or  by 
measuring  the  length,  I,  of  a  pendulum  which  will  vibrate  seconds. 
Extensive  surveys  of  this  sort  have  been  made,  and  are  still  in  prog- 
ress, and  it  is  found  that  the  force  of  gravity  at  the  pole  exceeds  that 
at  the  equator  by  about  ^^  part.  In  other  words,  a  person  who 


110  THE   EARTH   AS   AN   ASTRONOMICAL   BODY. 

weighs  190  pounds  at  the  equator  (by  a  spring  balance)  would,  if 
carried  to  the  pole,  show  191  pounds  by  the  same  balance. 

The  apparatus  most  used  at  present  for  the  purpose  of  measuring  the 
force  of  gravity  is  a  modification  of  the  so-called  Eater's  pendulum.  The 
pendulum  itself  now  usually  employed,  as  constructed  by  Repsold,  consists 
of  a  brass  tube  about  three  inches  in  diameter  and  about  four  feet  long : 
the  two  ends  are  alike  in  form,  but  one  end  is  weighted  and  the  other  is 
light.  Two  parallel  knife  edges  are  inserted  through  the  rod  at  right  angles, 
one  near  the  heavy  end  and  the  other  at  just  the  same  distance  from  the 
lighter  one,  and  the  weights  and  dimensions  .of  the  apparatus  are  so  adjusted 
that  the  time  of  vibration  will  be  very  approximately  the  same  whether  the  pendu- 
lum is  swung  heavy  end  up  or  light  end  up,  and  will  be  not  far  from  one  second. 
The  distance  between  the  knife  edges  will  then,  according  to  the  theory  of 
the  pendulum,  be  very  nearly  equal  to  the  length  of  a  simple  pendulum 
vibrating  in  the  same  time ;  and  the  small  difference  can  be  accurately  cal- 
culated when  we  know  the  exact  time  of  vibration,  each  end  up.  The  knife 
edges  swing  on  agate  planes  which  are  fastened  upon  a  firm  support ;  and 
great  pains  must  be  taken  to  have  the  support  really  firm.  Professor  Peirce 
of  our  Coast  Survey  a  few  years  ago  detected  important  errors  in  a  majority 
of  the  earlier  pendulum  observations,  due  to  insufficient  care  in  this  respect. 

152*.  In  1891  Professor  Mendenhall,  then  superintendent  of  the  United 
States  Coast  Survey,  greatly  improved  the  apparatus  by  substituting  for  the 
seconds  pendulum  a  half-seconds  one,  and  enclosing  it  in  a  tight  case  ex- 
hausted of  air.  This  renders  the  instrument  much  more  manageable  and 
portable,  and  avoids  almost  entirely  the  troublesome  and  uncertain  correc- 
tion for  the  resistance  of  the  air.  Two  little  mirrors,  one  attached  to  the 
pendulum  itself  and  the  other  fixed  near  it  in  the  case,  give  the  means  of 
observing  the  pendulum  swing  by  watching  the  reflection  of  a  flash  produced 
electrically  every  second  by  the  clock  or  chronometer  which  furnishes  the 
time.  With  this  apparatus  the  determinations,  however,  are  merely  relative, 
the  pendulum  being  used  simply  as  "invariable,"  without  inversion.  A 
somewhat  similar,  but  less  elaborate  arrangement,  with  a  half-seconds  pen- 
dulum, was  still  earlier  introduced  in  Europe  by  Von  Sterneck. 

153.  The  observations  consist  in  comparing  the  pendulum  with  a  clock, 
either  by  noting  the  "  coincidences"  or  by  an  electrical  record  automatically 
made  on  a  chronograph.  The  observations  need  to  be  carefully  corrected 
for  temperature  (which,  of  course,  affects  the  distance  between  the  knife 
edges),  for  the  length  of  arc  through  which  the  pendulum  is  swinging,  and 
for  the  resistance  of  the  air.  The  observations  determine  the  "  force  of  grav- 
ity" (French  "pesanteur")  at  the  station.  This  "force  of  gravity,"  how- 
ever, thus  determined,  is  not  simply  the  earth's  attraction,  but  includes  also 
the  effects  of  the  centrifugal  force,  due  to  the  earth's  rotation,  which  we 
must  consider  and  allow  for. 


THE  EARTH'S  CENTRIFUGAL  FORCE.  Ill 

EFFECT  OF  CENTRIFUGAL  FORCE  DUE  TO  EARTH'S  ROTATION. 

154.     At  the  equator  the  centrifugal  force  acts  vertically  in  direct 
opposition  to  gravity,  and  is  given  by  the  well-known  formula 


(see  Physics,  p.  17),  in  which  Fis  the  velocity  of  the  earth's  sur- 
face at  the  equator,  and  R  the  earth's  radius.  Since  V  is  equal  to 
the  earth's  circumference  divided  by  the  number  of  seconds  in  a 
sidereal  day,  we  have 

V= ,  and  C '  = . 

t  £2 

Now  R,  the  radius  of  the  earth,  equals  20,926,000  feet ;  and  t  equals 
86,164  mean-time  seconds.  (7,  therefore,  comes  out  0.111  feet,  which 
is  -2\-§  of  g,  g  being  32£  feet. 

We  may  remark  in  passing  that  if  the  rate  of  rotation  were  seventeen 
times  as  great,  C  would  be  172,  or  289  times  greater  than  now,  and  would 
equal  gravity  ;  so  that  on  that  supposition  bodies  at  the  equator  would 
weigh  absolutely  nothing,  and  any  greater  velocity  of  rotation  would  send 
them  flying. 

At  any  other  latitude,  since  MN=  OQ  cos  MOQ,1  the  centrifugal 
force,  c,  equals  C  cos  <£,  acting  at  right  angles  to  the  axis  of  the  earthv^ 
and  parallel  to  the  plane  of  the  equator.     Now,  this  centrifugal 
c  is  not  wholly  effective  in  diminishing 
the  weight  of  a  body,  but  only  that  por- 
tion of    c    (MR   in  Fig.  52)  which  is 
directed  vertically.      c  is  MT  in   the 
figure,  and  MR  is  equal  to  c  multiplied 
by  the  cosine  of  <£,  which  finally  gives 
us  C  X  cos2<£  for  the  amount  by  which 
the  centrifugal  force  diminishes  gravity  ^ 

at  a  station  whose  latitude  is  <£.  FlG  52 

Every  determination,  therefore,  of  the        The  Earth's  Centrifugal  Force, 
"  force  of  gravity,"  obtained  by  the  pendulum,  needs  to  be  increased 
by  the  quantity 

y    vx  ~~~  2  i 


1  This  is  not  exact,  since  MNin  an  oblate  spheroid  is  less  than  OQ  X  cos  MOQ; 
but  the  difference  is  unimportant  in  the  case  of  the  earth. 


112  THE   EARTH   AS   AN   ASTRONOMICAL   BODY. 

in  order  to  get  the  real  value  of  the  earth's  gravitational  attraction 
at  the  point  of  observation. 

The  other  component  of  c  (viz.  MS)  acts  at  right  angles  to  gravity  and 
parallel  to  the  earth's  surface,  and  is  given  by  the  formula 

C  cos  <f>  sin  <j>  =  %  C  sin  2  <f>. 

The  direction  of  still  water  is  determined  by  the  resultant  of  the  earth's 
attraction  combined  with  this  deflecting  force  acting  towards  the  equator  ; 
so  that  this  surface  is  not  perpendicular  to  a  line  drawn  towards  the  centre 
of  the  earth  anywhere  excepting  at  the  equator  and  the  poles. 

155.  Having  a  series  of  pendulum  observations,  we  can  then  form 
a  table  showing  the  force  of  gravity  at  each  station ;  and  correcting 
this  by  adding  the  amount  of  the  centrifugal  force  at  each  place,  we 
shall  have  the  force  of  the  earth's  attraction.  This  is  greater  the 
nearer  each  station  is  to  the  centre  of  the  earth  ;  but  unfortunately 
there  is  no  simple  relation  connecting  the  force  with  the  distance. 
The  attraction  depends  not  only  on  the  distance  from  the  centre  of 
the  earth,  but  also  upon  the  form  of  the  earth  and  the  constitution 
of  its  interior,  and  the  arrangement  of  its  strata  of  different  density. 
We  may  safely  assume,  however,  that  the  earth  is  made  up  concen- 
trically, so  to  speak ;  the  strata  of  equal  density  being  arranged  like 
the  coats  of  an  onion.  On  this  hypothesis  Clairaut,  in  1742,  demon- 
strated the  relation  given  below,  which  is  always  referred  to  as 
Clairaut' s  equation. 

Let  w  be  the  loss  of  weight  between  the  equator  and  the  pole, 
and  C  the  centrifugal  force  at  the  planet's  equator,  both  being  ex- 
pressed as  fractions  of  the  equatorial  force  of  gravity,  and  let  d  be 
the  ellipticity  of  the  planet. 

Then,  as  Clairaut  proved, 

d  +  w  =  2$X  C; 
whence 


In  the  case  of  the  earth, 

1 


whence 
which  gives 


292.8* 


DIFFERENT   KINDS    OF    LATITUDE.  113 

But  the  different  results  obtained  from  pendulum  observations 
range  all  the  way  from  ^^  to 


155*.  As  regards  the  purely  astronomical  methods,  the  one  which 
depends  on  precession  and  nutation  requires  assumptions  respecting  the 
distribution  of  matter  within  the  earth  which  render  the  result  somewhat 
uncertain.  Harkness  deduces  by  it  a  value  of  ^y. 

The  lunar  perturbation  from  which  the  oblateness  of  the  earth  can  be 
calculated  is  very  small  (only  about  8"),  and  hardly  well  enough  determined 
as  yet.  According  to  Harkness  the  values  obtained  from  it  range  between 

*fo  and  STY- 

The  observations  of  star-occultations  during  lunar  eclipses  are  not  yet 

sufficiently  numerous  to  furnish  a  reliable  value. 

Considering  all  the  data  it  can  only  be  said  that  the  oblateness  probably 
lies  between  ^^  and  ^1^,  and  probably  nearer  the  latter  limit  than  the 
other.  Harkness,  in  his  "adjusted"  system  of  astronomical  constants,  gives 
as  his  final  result  1 


300.2±3.0 


156.    Astronomical,  Geographical,  and  Geocentric  Latitudes. — 

The  astronomical  latitude  of  a  place  has  been  denned  as  the  elevation 
of  the  pole,  or,  what  conies  to  the  same  thing,  it  is  the  angle  between 
the  plane  of  the  equator  and  the  direction  of  gravity  at  that  place, 
however  that  direction  may  be  affected  by  local  causes. 

The  geocentric  latitude,  on  the  other  hand,  is  the  angle  made  at 
the  centre  of  the  earth  (as  the  word  im- 
plies) between  the  plane  of  the  equator 
and  a  line  drawn  from  the  observer  to  the 
centre  of  the  earth,  which  line  of  course 
does  not  coincide  with  the  direction  of 
gravity,  since  the  earth  rotates,  and  is  not 
spherical. 

The  geographical  or  geodetic  latitude  of 
a  station  is  the  angle  formed  with  the  plane 
of  the  equator  by  a  line  drawn  from  the  FIG.  53. 

station  perpendicular  to  the  surface  of  the     Astronomical  and  Geocentric 

Latitude. 
standard  spheroid. 

If  the  earth's  surface  were  strictly  spheroidal,  and  there  were  no  local  varia- 
tions of  gravity,  the  astronomical  latitude  and  the  geographical  latitude 
would  coincide  —  and  they  never  differ  greatly  ;  but  the  geocentric  latitude 
differs  from  them  by  a  very  considerable  quantity  —  as  much  as  11'  in 
latitude  45°.  The  geocentric  latitude  is  but  little  used  except  in  certain 
astronomical  calculations  where  parallax  is  involved. 


114  DIFFERENT   KINDS   OF  LATITUDE. 

In  Fig.  53,  the  angle  MOQ  is  the  geocentric  latitude  of  M,  while  MNQ  is 
the  geographical  latitude.  MNQ  is  also  the  astronomical  latitude,  unless 
there  is  some  local  disturbance  of  the  direction  of  gravity.  The  angle  OMN, 
which  is  the  difference  between  the  geocentric  and  astronomical  latitudes,  is 
called  "  the  angle  of  the  vertical." 

157.  It  will  be  noticed  that  the  astronomical  latitude  of  a  place  is 
the  only  one  of  these  three  latitudes  which  is  determined  directly  by 
observation.     In  order  to  know  the  geocentric  and  geographical  lati- 
tudes of  a  place,  we  must  know  the  form  and  dimensions  of  the  earth, 
which  are  ascertained  only  by  the  help  of  observations  made  elsewhere. 

The  geocentric  degrees  are  longer  near  the  equator  than  near  the 
poles,  and  it  is  worth  noticing  that  if  we  form  a  table  giving  the  length 
of  each  degree  of  geographical  latitude  from  the  equator  to  the  pole, 
the  same  table,  read  backwards,  gives  the  length  of  geocentric  degrees. 

Since  the  earth  is  ellipsoidal  instead  of  spherical,  it  is  evident  that 
lines  of  "  level "  on  the  earth's  surface  are  affected  by  the  earth's  ro- 
tation. If  this  rotation  were  to  cease,  the  direction  of  gravity  would 
be  so  much  changed  that  the  Gulf  of  Mexico  would  run  up  the  Mis- 
sissippi River,  because  the  distance  from  the  centre  of  the  earth  to 
the  head  of  the  river  is  less  by  some  thousands  of  feet  than  the 
distance  from  the  mouth  of  the  river  to  the  centre  of  the  earth. 

158.  Station  Errors.  —  The  irregularities  in  the  direction  of  gravity 
are  by  no  means  insensible  as  compared  with  the  accuracy  of  modern  astro- 
nomical observation,  and  the  difference  between  the  astronomical  latitude 
and  longitude  of  a  place  and  the  geographical  latitude  and  longitude  of  the 
same  place  constitute  what  is  called  the  "station  error."     In  the  eastern  part 
of  the  United  States  these  station  errors,  according  to  the  Coast  Survey 
observations,  average  about  1J".     Errors  of  from  4"  to  6"  are  not  uncom- 
mon, and  in  mountainous  countries,  as  for  instance  in  the  Caucasus  and  in 
Northern  India,  these  errors  occasionally  amount  to  30"  or  40".     They  are 
not  "  errors  "  in  the  sense  that  the  astronomical  latitude  of  the  place  has  not 
been  determined  correctly,  but  are  merely  the  effects  of  the  irregular  distri- 
bution of  matter  in  the  crust  of  the  earth  in  altering  the  direction  of  gravity. 
Pendulum  observations  show  local  variations  in  the  force  of  gravity  quite 
proportional  to  the  deviations  which  the  station-errors  show  in  its  direction. 

IV. 

159.  The  Earth's  Mass  and  Density.  — The  <  mass '  of  a  body  is  the 
quantity  of  matter  that  it  contains,  the  unit  of  mass  being  the  quantity 
of  matter  contained  in  a  certain  arbitrary  body  which  is  taken  as  a 
standard.     For  instance,  a  "kilogram"  is  the  quantity  of  matter 


GRAVITATION.  115 

contained  in  the  block  of  platinum  preserved  at  Paris  as  the  stand- 
ard of  mass.  A  pound  is  similarly  defined  by  reference  to  the  pro- 
totypes at  Washington  and  London. 

Two  masses  of  matter  are  denned  as  equal  which  require  the  same 
expenditure  of  energy  to  give  them  the  same  velocity;  or,  vice  versa, 
those  are  equal  which,  when  they  have  the  same  velocity,  possess  the 
same  energy,  and,  in  giving  up  their  motion  and  coming  to  rest,  do  the 
same  amount  of  work  (i.e.,  they  have  the  same  "inertia"). 

Masses  can  therefore  be  compared  by  subjecting  them  to  the  action  of 
some  given  force  (stress),  and  comparing  the  energies  developed  in  them 
when  they  have  moved  equal  distances,  or  the  velocities  attained  at  the  end 
of  a  given  time. 

160.  Proportionality  of  Mass  to  Weight.  —  Newton  showed  by 
his  experiments  with  pendulums  of  different  substances,  that  at  any 
given  point  the  attraction  of  the  earth  for  a  body  of  any  kind  of 
matter  is  proportional  to  the  mass  of  that  body  ;  the  attraction  be- 
ing measured  as  a  pull  or  "  stress  "  in  this  case,  and  called  u  the 
weight"  of  the  body.  In  other  and  more  common  language,  the 
mass  of  a  body  is  proportional  to  its  weight  (we  must  not  say  it  is  its 
weight),  provided  the  weighing  of  the  bodies  thus  compared  is  done, 
in  cases  where  scientific  accuracy  is  essential,  at  the  same  place  on 
the  earth's  surface.  Practically,  therefore,  we  usually  measure  the 
masses  of  bodies  by  simply  weighing  them.1  It  is  to  be  carefully  ob- 
served, however,  that  the  words  "  kilogram,"  "  pound/7  "  ton,"  etc., 
have  also  a  secondary  meaning,  as  denoting  units  of  pull  and  push, 
—  of  "stress,"  speaking  strictly  and  technically,  —  or  of  "force," 
as  that  much  abused  word  is  very  generally  used. 

It  is,  from  a  literary  point  of  view,  just  as  proper  to  speak  of  a  stress  or  a 
pull  of  a  hundred  pounds2  as  of  a  mass  of  a  hundred  pounds,  but  the  word 
"  pound  "  means  an  entirely  different  thing  in  the  two  cases.  At  the  sur- 
face of  the  earth  the  relation  between  the  ideas,  however,  is  so  close  that 
the  way  in  which  the  ambiguity  came  about  is  perfectly  obvious,  and  it  is 
hardly  probable  that  language  will  ever  change  so  as  to  remove  it.  To  a 
certain  extent  it  is  admittedly  unfortunate,  and  the  student  must  always  be 
on  his  guard  against  it.  At  the  earth's  surface  a  mass  of  100  pounds  always 
"  weighs  "  very  nearly  100  pounds  ;  but,  to  anticipate  slightly,  at  an  elevation 


1  See  note  at  the  end  of  chapter,  page  123. 

2  The  scientific  and  unambiguous  unit  of  stress  is  the  dyne,   which  equals 
the  weight  at  Paris  of  ^g^¥  of  a  gram,  —  nearly  1.02  milligrams. 


116  THE   EARTH   AS   AN   ASTRONOMICAL   BODY. 

of  4000  miles  above  the  surface,  the  same  mass  would  "  weigh  "  only  25 
pounds  ;  at  the  distance  of  the  moon  about  half  an  ounce  ;  while  on  the 
surface  of  the  sun  it  would  "  weigh  "  nearly  2800  pounds  (of  stress). 

161.  Gravity,  —  The  law  of  gravitation  discovered  by  Newton 
declares  that  any  particle  of  matter  attracts  any  other  particle  with  a 
force  ("  stress,"  if  the  bodies  are  prevented  from  moving)  proportional 
inversely  to  the  square  of  the  distance  between  them,  and  directly  to 
the  product  of  their  masses  ;  or,  as  a  formula,  we  may  write, 


in  which  M1  and  Mz  are  the  two  masses,  and  d  the  distance  between 
them,  while  G  is  a  constant  numerical  factor  which  depends  upon 
the  system  of  units  employed. 

It  is  known  as  the  "  Newtonian  Constant  "  or  the  "  Constant  of  Gravita- 
tion," being  supposed  to  maintain  the  same  value  throughout  the  universe. 
According  to  the  most  recent  determination,  —  that  of  Boys  in  1893 
(Art.  166),  —  its  value  in  the  C.  G.  S.  system  (centimetre-gram-second)  is 
666  x  1O~10  dynes  ;  i.e.,  two  balls,  each  having  a  mass  of  one  gram,  and  with 
their  centres  one  centimetre  apart,  would  attract  each  other  with  a  force 
(stress)  of  666  ten-thousand-millionths  of  a  dyne.  txOC'MHXv  " 

The  "  acceleration  "  of  a  particle  due  to  the  attraction  of  a  mass,  M,  at 

M 

distance,  d,  is  given  by  the  equation,  /=  G'  —  ;  and  when  two  masses,  M^ 

and  M2  (which  are  free  to  move)  attract  each  other  their  relative  acceleration 
is  the  sum  of  the  two  accelerations  which  each  produces  in  the  other.  It  is 

therefore  given  by  the  formula,  /=  G'  —  l—^  —  -.    (Note  that  in  this  we  have 

the  sum  of  the  masses,  instead  of  their  product.)  In  the  C.  G.  S.  system  G' 
is  numerically  identical  with  G,f  being  measured  in  cm.  per  sec. 

We  must  not  imagine  the  word  "  attract  "  to  mean  too  much.  It  merely 
states  the  fact  that  there  is  a  tendency  for  the  bodies  to  move  toward  each 
other,  without  including  or  implying  any  explanation  of  the  fact.  So  far, 
no  explanation  has  appeared  which  is  less  difficult  to  comprehend  than  the 
fact  itself. 

162.  When  the  distance  between  attracting  bodies  is  large  as  com- 
pared with  their  own  magnitude,  then,  reckoning  the  distance  between 
their  centres  of  mass  as  their  true  distance,  the  formula  is  sensibly 
true  for  them  as  it  would  be  for  mere  particles.  When,  however,  the 
distance  is  not  thus  great,  the  calculation  of  the  attraction  becomes  a 
very  serious  problem,  involving  what  is  known  as  a  "  double  integra- 


MASS  AND  DENSITY  OF  THE  EARTH.         117 

tion."  We  must  find  the  attraction  of  each  particle  of  the  first  body 
upon  each  particle  of  the  other  body,  and  take  the  sum  of  all  these 
infinitesimal  stresses.  Newton,  however,  showed  that  if  the  bodies 
are  spheres,  either  homogeneous  or  of  concentric  structure,  then  they 
attract  and  are  attracted  precisely  as  if  the  matter  in  them  were  wholly 
collected  at  their  centres.  The  earth,  for  instance,  attracts  a  body  at 
its  surface  very  nearly  as  if  it  were  all  collected  at  its  own  centre, 
4000  miles  distant ;  not  exactly  so,  because  the  earth  is  not  strictly 
spherical ;  but  in  what  follows  we  shall  neglect  this  slight  inaccuracy. 

163,  In  order,  then,  to  find  the  mass  of  the  earth  in  kilograms, 
pounds,  or  tons,  we  must  find  some  means  of  accurately  comparing 
its  attraction  for  some  object  on  its  own  surface  with  the  attraction 
of  the  same  object  by  some  body  of  known  mass,  at  a  measured  dis- 
tance.    The  difficulty  lies  in  the  fact  that  the  attraction  produced 
by  any  body,  not  too  large  to  be  handled  conveniently,  is  so  exces- 
sively small  that  only  the  most  delicate  operations  serve  to  detect 
and  measure  it. 

The  first  successful  attack  upon  the  problem  was  made  in  1774  by 
Maskelyne,  the  Astronomer  Eoyal,  by  means  of  what  is  now  usually 
referred  to  as  :  — 

164.  1.  "  THE  MOUNTAIN  METHOD,"  because,  in  fact,  the  earth 
in  this  operation  is  weighed  against  a  mountain. 

Two  stations  were  chosen  on  the  same  meridian,  one  north  and  one 
south  of  the  mountain  Schehallien,  in  Scotland.  In  the  first  place, 
a  careful  topographical  survey  was 

made  of  the  whole  region,  giving                 VK      ^/^~\^       N't 
the  precise  distance  between  the  sta- 
tions, as  well  as  the  exact  dimensions  -A  

of  the  mountain,  which  is  a  "hog-    — jf  J?^ — -* 

back"  of  very  regular  contour.  From  FlG>  M' 

, ,       ,  -. .  .  „    , ,  . ,         The  Mountain  Method  of  Determining 

the  known  dimensions  of  the  earth  the  Earth»s  Density. 

and  the  measured  distance,  the  dif- 
ference of  the  geographical  latitudes  of  the  two  places  M  and  jV(Fig. 
54)  can  be  accurately  computed  ;  i.e.,  the  angle  which  the  plumb 
lines  at  M  and  N  would  have  made  if  there  were  no  mountain  there. 
In  this  case  it  was  41".  JThenext  operation  was  to  observe  the 
astronomical  latitude  at  each  station.  This  astronomical  difference 
"^oTTatitude,  i.e.,  the  angle  which  the  plumb  lines  actually  do  make, 
was  found  to  be  53",  the  plumb  lines  at  M  and  N  being  drawn  inward 
out  of  their  normal  position  by  the  attraction  of  the  mountain  to  the 


118 


THE   EAKTH   AS   AN   ASTRONOMICAL   BODY. 


extent  of  6"  on  each  side  ;  so  that  the  astronomical  difference  of 
latitude  was  increased  by  12"  over  the  geographical. 


Now,  in  such  a  case  the  ratio  of  gravity  to  the  deflecting  force, 
according  to  the  laws  of  the  composition  of  forces,  is  that  of 
aM  to  aA'  in  the  figure  (Fig.  55),  or  the  ratio  of  1  to  the  tan- 
gent  of  the  deflection,  8  ;  that  is,  calling  the  deflecting  force  /, 

we  have  —  =  cot  8,  =  cot  6"  in  this  case. 

By  the  law  of  gravitation,  the  earth's  attracting  force  at  its 
surface  is  given  by  the  formula 


where  E  is  the  mass  of  the  earth  (the  unknown  quantity  of  our  problem), 
and  R  its  radius,  4000  miles.  Similarly,  if  C  in  the  figure  is  the  centre  of 
attraction  of  the  mountain,  we  have 


m  being  the  mass  of  the  mountain,  and  d  the  distance  from  C  to  the  station. 
Combining  this  with  the  preceding,  we  get 


m 


We  thus  get  the  ratio  of  the  earth's  mass  to  that  of  the  mountain; 
and  provided  we  can  find  the  mass  of  the  mountain  in  tons  or  any 
other  known  unit  of  mass,  the  problem  will  be  completely  solved. 
By  a  careful  geological  survey  of  the  mountain,  with  deep  borings 
into  its  strata,  the  mass  of  the  mountain  was  determined  as  accu- 
rately as  it  could  be  (though  here  is  the  weakest  point  of  the  method), 
and  thus  the  mass  of  the  earth  was  finally  computed. 

Now,  knowing  the  diameter  of  the  earth,  its  volume  in  cubic  feet 
is  easily  found,  and  from  the  volume  and  the  known  number  of 
mass-pounds  (62£  nearly)  in  a  cubic  foot  of  water,  the  weight  the 
earth  would  have,  if  composed  of  water,  follows.  Comparing  this 
with  the  mass  actually  found,  we  get  the  density,  which  in  this  ex- 
periment came  out  4.71. 

A  repetition  of  the  work  in  1832  at  Arthur's  Seat,  near  Edin- 
burgh, gave  5.32,  and  several  later  determinations  have  since  been 
made  by  this  method,  giving  results  in  near  agreement  with  that 
stated  in  Art.  132. 


MASS  AND  DENSITY  OF  THE  EARTH. 


119 


165.  2.  Much  more  trustworthy  results,  however,  are  obtained 
by  the  method  of  the  TORSION  BALANCE,  first  devised  by  Michell, 
but  first  employed  by  Cavendish  in  1798.  A  light  rod,  carrying  two 
small  balls  at  its  extremities,  is  suspended  horizontally  at  its  centre 
by  a  long  fine  metallic  wire.  If  it  be 
allowed  to  come  to  rest,  and  then  a 
very  slight  deflecting  force  be  applied, 
the  rod  will  be  pulled  out  of  position 
by  an  amount  depending  on  the  stiff- 
ness and  length  of  the  wire,  as  well 
as  the  force  itself.  When  the  deflect- 
ing force  is  removed,  the  rod  will 
vibrate  back  and  forth  until  brought 
to  rest  by  the  resistance  of  the  air. 
The  "torsional  coefficient,"  as  it  is 
called  (i.e.,  the  stress  corresponding 
to  a  torsion  of  one  revolution),  can  be 
accurately  determined  by  observing 
the  time  of  vibration  when  the  dimen- 
sions and  weight  of  the  rod  and  balls 
are  known.  If,  now,  two  large  balls 
A  and  B  are  brought  near  the  smaller 
ones,  as  in  Fig.  56,  a  deflection  will  be  produced  by  their  attraction, 
and  the  small  balls  will  move  from  a  and  b  to  a)  and  b1.  By  shift- 
ing the  large  balls  to  the  other  side  at  A'  and  B',  we  get  an  equal 
deflection  in  the  opposite  direction,  i.e.,  to  a1'  and  b",  and  the  differ- 
ence between  the  two  positions  assumed  by  the  small  balls,  i.e., 
a'a"  and  b'b",  will  be  twice  the  deflection. 

It  is  not  necessary,  nor  even  best,  to  wait  for  the  balls  to  come  to  rest. 
We  note  the  extremities  of  their  swing.  The  middle  point  of  the  swing 
gives  the  point  of  rest,  and  the  time  occupied  by  the  swing  is  the  time  of 
vibration,  which  we  need  in  determining  the  coefficient  of  torsion.  We 
must  also  measure  accurately  the  distance,  Aa'  and  jB6',  between  the  centre 
of  each  of  the  large  baUs  and  the  point  of  rest  of  the  small  ball  when  deflected. 


Ar 


\ 


FIG.  56.  —  Plan  of  the  Torsion  Balance. 


The  earth's  attraction  on  each  of  the  small  balls  of  course  equals 
the  ball's  weight.  The  attractive  force  of  the  large  ball  on  the  small 
one  near  it  is  found  directly  from  the  experiment.  If  the  deflection, 
for  instance,  is  1°  and  the  coefficient  of  torsion  is  such  that  it  takes 
one  grain  to  twist  the  wire  around  one  whole  revolution,  then  the 
deflecting  force,  which  we  will  call  /  as  before,  will  be  ^^  of  a  grain. 


120  THE  EARTH   AS   AN    ASTRONOMICAL   BODY. 

Call  the  mass  of  the  large  ball  B,  and  let  d  be  the  measured  distance 
from  its  centre  to  that  of  the  deflected  ball.     We  shall  then  have 


also,  w  being  the  weight  of  the  small  ball, 


whence  we  get,  very  much  as  in  the  preceding  case, 


The  method  differs  from  the  preceding  in  that  we  use  a  large  ball 
of  metal  instead  of  a  mountain,  and  measure  its  deflecting  force  by 
a  laboratory  experiment  instead  of  comparing  astronomical  observa- 
tions with  geodetic  measurements. 

166.  In  the  earlier  experiments  by  this  method  the  small  balls  were  of 
lead,  about  two  inches  in  diameter,  at  the  extremities  of  a  light  wooden  rod, 
five  or  six  feet  long,  enclosed  in  a  case  with  glass  ends,  and  their  position 
and  vibration  was  observed  by  a  telescope  looking  directly  at  them  from  a 
distance  of  several  feet.  The  attracting  masses,  B,  were  balls  also  of  lead, 
about  one  foot  in  diameter,  mounted  on  a  frame  pivoted  in  such  a  way  that 
they  could  be  easily  brought  to  the  required  positions. 

Great  difficulty  was  caused  by  air  currents  in  the  case,  and  it  was  neces- 
sary to  enclose  the  whole  apparatus  in  a  small  room  of  its  own  which  was 
covered  with  tin-foil  on  the  outside,  and  to  avoid  going  near  the  room  or 
allowing  any  radiant  heat  to  strike  it  for  hours  before  the  observations. 
Baily,  in  England,  and  Reich,  in  Germany,  between  1838  and  1842,  made 
very  extensive  series  of  observations  of  this  kind.  Baily  obtained  5.66  for 
the  earth's  density,  and  Reich  5.48. 

The  experiment  was  repeated  in  1872  by  Cornu,  in  Paris,  with  a  modified 
apparatus. 

The  horizontal  bar  was  in  this  case  only  half  a  metre  long,  of  aluminium, 
with  small  platinum  balls  at  the  end.  For  the  large  balls,  glass  globes  were 
used,  which  could  be  pumped  full  of  mercury  or  emptied  at  pleasure.  The 
whole  was  enclosed  in  an  air-tight  case,  and  the  air  exhausted  by  an  air- 
pump.  The  deflections  and  vibrations  were  observed  by  means  of  a  tele- 

1  Note  that  G  is  directly  given  by  the  observations  ;  G  =/—  . 


MASS  AND  DENSITY  OF  THE  EARTH.         121 

scope  watching  the  image  of  a  scale  reflected  in  a  small  mirror  attached  to 
the  aluminium  beam  near  its  centre,  according  to  the  method  first  devised 
by  Gauss  and  now  so  generally  used  in  galvanometers  and  similar  apparatus. 
Cornu  obtained  5.56  as  the  result,  and  showed  that  Baily's  figure  required  a 
correction  which,  when  applied,  would  reduce  it  to  5.55. 

A  still  more  recent  and  elaborate  repetition  of  the  experiment  was  made 
by  Boys  at  Oxford  in  1890-1893.  The  beam,  only  about  half  an  inch  long, 
was  suspended  in  a  partial  vacuum  by  a  torsion-fibre  of  quartz.  The  at- 
tracted balls  were  of  gold,  a  quarter  of  an  inch  in  diameter,  and  the  attracting 
balls  were  of  lead,  41  and  2^  inches  in  diameter  —  two  sets.  His  result  for 
the  density  of  the  earth  was  5.527.  Still  more  recently  (in  1897)  Braun  of 
Mariaschein  (Bohemia)  publishes  a  result  obtained  by  the  same  general 
method  and  in  perfect  agreement  with  that  of  Boys. 

167.  3.  POTSDAM  OBSERVATIONS. — During  1886  and  1887  an- 
other series  of  observations  was  made  by  Wilsing,  at  Potsdam,  with 
apparatus  similar  in  principle  to  the  torsion  balance,  except  that 
the  bar  carrying  the  balls  to  be  attracted  was  vertical,  and  turned 
on  knife  edges  very  near  its  centre  of  gravity.     The  knife  edges, 
like  those  of  an  ordinary  balance,  rested  upon  agate  planes,  and  the 
centre  of  gravity  of  the  apparatus  was  so  adjusted  that  one  vibration 
of  the  pendulum,  under  the  influence  of  gravity  alone,  would  occupy 
from  two  to  four  minutes.     The  deflecting  weights  in  this  case  were 
large  cylinders  of  cast  iron,  suspended  in  such  a  way  that  they 
could  be  brought  opposite  the  small  balls,  first  on  one  side  and  then 
on  the  other.     The  whole  was  set  up  in  a  basement,  and  carefully 
and  very  effectually  guarded  against  all  changes  of  temperature,  the 
arrangements  being  such  that  all  manipulations  and  observations 
could  be  effected  from  the  outside  without  entering  the  room.     The 
deflections  and  vibrations  were  observed  by  a  reflected  scale,  as  in 
Cornu's  observations.    The  result  obtained  was  5.59.     Several  other 
methods  have  been  used  ;  of  less  scientific  value,  however. 

168.  «•     The  mass  of  the  earth  can  be  deduced  by  ascertaining  the  force 
of  gravity  at  the  top  of  a  mountain  and  at  its  base,  by  means  of  pendulum  experi- 
ments.    The  mass  of  the  mountain  must  be  determined  by  a  survey,  just  as 
in  the  Schehallien  method,  which  makes  the  method  unsatisfactory.     At  the 
top  of  a  mountain  the  height  of  which  is  h,  and  the  distance  of  its  centre  of 
attraction  from  the  top  is  d,  gravity  will  be  made  up  of  two  parts,  one  the 
attraction  of  the  earth  at  a  distance  from  its  centre  equal  to  R  +  h,  and  the 
other  the  attraction  of  the  mountain  alone  considered.     Calling  the  mass  of 
the  mountain  m,  and  gravity  at  its  summit  g'  (g  being  the  force  of  gravity  at 
the  earth's  surface),  we  shall  have  the  proportion 


E     r       E-          m 

'' 


"L 


122  THE   EAETH  AS   AN   ASTRONOMICAL  BODY. 

the  second  fraction  in  the  last  term  of  the  proportion  being  the  attraction 
of  the  mountain.  When  g  and  g1  are  ascertained  by  the  pendulum  experi- 
ments, E  remains  as  the  only  unknown  quantity,  and  can  be  readily  found. 
Observations  of  this  kind  were  made  by  Carlini,  in  1821,  on  Mt.  Cenis,  and 
the  result  was  4.95 :  also  by  Mendenhall  on  Fusiyama  in  1890,  and  by 
Preston  on  Mauna  Kea  in  1892,  the  results  being  respectively  5.77  and  5.57. 

169.  b.    By  means  of  pendulum  observations  at  the  earth's  surface  compared 
with  those  at  the  bottom  of  a  mine  of  known  depth.     This  method  was  employed 
by  Airy  in  1854,  at  Harton  Colliery,  1200  feet  deep  ;  result,  6.56.     In  this 
case  the  principle  involved  is  somewhat  different.     At  any  point  within  a 
hollow,  homogeneous,  spherical  shell,  gravity  is  zero,  as  Newton  has  shown. 
The  attraction  balances  in  all  directions.     If,  then,  we  go  down  into  a  mine, 
the  effect  on  gravity  is  the  same  as  if  a  shell  composed  of  all  that  part  of  the 
earth  above  our  level  had  been  removed.     At  the  same  time  our  distance 
from  the  earth's  centre  has  been  decreased  by  dt  the  depth  of  the  mine. 

At  the  surface  g  —  G j^ ,  as  before. 

At  the  bottom  of  the  mine  gf  =  G — -r-j- . 

(it  —  a) 

Comparing  the  two  equations,  we  find  E  in  the  terms  of  the  shell,  since 
the  ratio  of  g  to  g'  is  given  by  pendulum  observations.  Obviously,  however, 
the  mass  of  the  "  shell "  is  difficult  to  determine  with  accuracy.  And  it  is 
by  no  means  homogeneous,  so  that  there  is  no  great  reason  for  surprise  at 
the  discordant  result.  g>  was  found  to  be  actually  greater  than  g,  showing 
that  although  at  the  centre  of  the  earth  the  attraction  necessarily  becomes  zero, 
yet  as  we  descend  below  the  surface,  gravity  increases  for  a  time  down  to  some 
unknown  but  probably  not  very  great  depth,  where  it  becomes  a  maximum. 

170.  c.   By  experiments  with  a  common  balance.     If  a  body  be  hung  from 
one  of  the  scale-pans  of  a  balance,  its  apparent  weight  will  obviously  be 
increased  when  a  large  body  is  brought  very  near  it  underneath  ;   and  this 
increase  can  be  measured.     Poynting  in  England  and  Jolly  in  Germany 
have  recently  used  this  method,  and  have  obtained  results  agreeing  very 
fairly  with  those  got  from  the  torsion  balance.     A  series  of  observations  by 
a  modification  of  this  method,  and  on  a  very  large  scale,  has  been  carried 
out  at  Berlin,  and  the  result  published  late  in  1896  by  Richarz,  is  5.505,  in 
excellent  accordance  with  Poynting  who  in  1891  got  5.493. 

171.  Constitution  of  the  Earth's  Interior.  —  Since  the  average 
density  of  the  earth's  crust  does  not  exceed  three  times  that  of  wa- 
ter, while  the  mean  density  of  the  whole  earth  is  about  5.53  (taking 
the  average  of  all  the  most  trustworthy  results),  it  is  obvious  that 
at  the  centre  the  density  must  be  very  much  greater  than  at  the 
surface,  —  very  likely  as  high  as  eight  or  ten  times  that  of  water, 


MASS  AND  DENSITY  OF  THE  EAKTH. 


123 


and  equal  to  the  density  of  the  heavier  metals.  There  is  nothing  in 
this  that  might  not  have  been  expected.  If  the  earth  were  ever 
fluid,  it  is  natural  to  suppose  that  in  the  solidification  the  densest 
materials  would  settle  towards  the  interior. 

Whether  the  interior  of  the  earth  is  solid  or  fluid  it  is  difficult  to  say  with 
certainty.  Certain  tidal  phenomena,  to  be  discussed  hereafter,  have  led  Sir 
William  Thomson  (now  Lord  Kelvin)  and  the  younger  Darwin  to  conclude 
that  the  earth  as  a  whole  is  solid  throughout,  and  "  more  rigid  than  steel," 
volcanic  centres  being  mere  pustules  in  the  general  mass.  To  this  many 
geologists  demur. 

As  regards  the  temperature  at  the  earth's  centre,  it  is  hardly  an  astro- 
nomical question,  though  it  has  very  important  astronomical  relations.  We 
can  only  take  space  to  say  that  the  temperature  appears  to  increase  from 
the  surface  downward  at  the  rate  of  about  one  degree  Fahrenheit  for  every 
fifty  or  sixty  feet,  so  that  at  the  depth  of  a  few  miles  the  temperature  must 
be  very  high. 


171  *  (Note  to  Art.  160.)  Measurement  of  Mass  by  Means  of  In- 
ertia. —  It  is  quite  possible  to  measure  masses  without  weighing.  In  Fig. 
126,  B  is  a  receptacle  carried  at  the  end  of  a  horizontal  arm  A,  which  is 
itself  attached  to  an  axis  MN,  exactly 
vertical  and  free  to  turn  on  pivots  at 
top  and  bottom.  A  spiral  spring  S, 
like  the  hair  spring  of  a  watch,  is  con- 
nected with  this  axis  so  that  if  A  is 
disturbed  it  will  oscillate  back  and 
forth  at  a  rate,  which  depends  upon 
the  stiffness  of  the  spring  and  the  total 
moment  of  inertia  of  the  apparatus. 
If  we  put  into  B  one  standard  "pound" 
(of  mass),  it  will  vibrate  a  certain 
number  of  times  a  minute  ;  if  two 
pounds,  it  will  vibrate  more  slowly;  if 
three,  still  more  slowly  ;  and  so  on : 
and  this  time  of  vibration  can  be  de- 
termined and  tabulated.  To  determine 
now  the  mass  of  a  body  X,  we  have 

only  to  put  it  into  the  receptacle  B,  set  the  apparatus  vibrating,  and  count 
the  number  of  swings  in  a  minute.  Referring  to  our  table,  we  find  what 
number  of  "  pounds  "  in  B  would  have  given  the  same  rate  of  vibration. 
We  know  then  that  the  "  inertia  "  of  X  is  the  same  as  that  of  this  number 
of  "  pounds,"  and  therefore  its  mass  is  the  same. 

This  determination  is  independent  of  all  considerations  of  weight:  the 
apparatus  would  give  the  same  results  on  the  surface  of  the  moon,  or  on  that 


FIG.  53* 


124  THE   EARTH   AS   AN   ASTRONOMICAL  BODY. 

of  Jupiter,  as  on  the  earth.  It  is  obvious,  however,  that  an  instrument  of 
this  sort  could  not  compete  in  accuracy  or  convenience  with  a  well-made 
balance,  because  of  the  friction  of  the  pivots,  the  resistance  of  the  air,  etc. 
We  introduce  it  simply  to  assist  in  separating  the  idea  of  mass  from  that  of 


EXERCISES  ON  CHAPTER  V. 

1.  Does   the   transportation   of   sediment   by  the  Mississippi  tend  to 
lengthen  or  to  shorten  the  day? 

2.  If  the  diameter  of  the  earth  were  doubled,  keeping  its  mass  unchanged, 
how  would  its  density  and  the  weight  of  bodies  at  its  surface  be  affected  ? 

3.  If  its  diameter  were  trebled,  keeping  its  density  unchanged,  how  much 
would  its  mass  and  the  weight  of  bodies  at  its  surface  be  increased  ? 

4.  Supposing  the  earth  to  be  homogeneous,  how  great  (approximately) 
would  be  the  force  of  gravity  a  thousand  miles  below  its  surface  ? 

5.  Assuming  the  earth  to  be  homogeneous,  at  what  depth  (approximately) 
would  a  pendulum  which  at  the  surface  of  the  earth  vibrates  seconds  vibrate 
in  a  second  and  a  quarter  ?  Ans.    1440  miles. 

6.  Given  two  spheres  one  of  which  has  a  mass  m  times  greater  than  the 
other  :  on  what  point  on  the  line  joining  their  centres  are  their  attractions 
equal? 

Solution.  Let  d  be  the  distance  between  their  centres,  and  x  the  distance 
of  the  point  of  equilibrium  from  the  smaller  body :  then  the  attraction  of 

the  larger  body  at  that  point  is  G  — -,  that  of  the  smaller  being  G  —^ 

(a—x)  x . 

Canceling  the  6?s,  and  taking  the  square  roots,  we  have  — =  -  ;   from 

which  we  have 

7.  Assuming  the  moon's  mass  as  ¥JT  of  the  earth's,  where  is  the  equilib- 
rium point  on  the  line  of  centres  ? 

Ans.    At  a  point  one-tenth  of  the  distance  from  the  moon  to  the  earth. 

8.  Assuming  the  distance  of  the  sun  as  93  000000  miles,  and  its  mass 
as  330000  times  that  of  the  earth,  where  is  the  point  at  which  their  attrac- 
tions balance  on  the  line  of  centres  ? 

93  000000        93  000000 

Ans.  Distance  from  the  earth  = =  -^TTITTT-  =  161600  miles. 

1+V330000        575.456 

Note.  —  This  distance  is  much  less  than  the  distance  of  the  moon  from  the  earth,  so  that 
at  the  time  of  new  moon  the  sun's  attraction  upon  the  moon  very  much  exceeds  that  of  the 
earth.  Compare  Art.  439. 


THE   ANNUAL   MOTION   OF   THE   SUN.  125 


CHAPTER  VI. 

THE  APPARENT  MOTION  OF  THE  SUN  AMONG  THE  STARS, 
AND  THE  EARTH'S  ORBITAL  MOTION.  —  THE  EQUATION 
OF  TIME,  PRECESSION,  NUTATION,  AND  ABERRATION.  — 


172,  The  Annual  Motion  of  the  Sun,  —  The  apparent  annual  mo- 
tion of  the  sun  must  have  been  one  of  the  earliest  noticed  of  all 
astronomical  phenomena.     Its  discovery  antedates  history. 

As  seen  by  the  people  in  Europe  and  Asia,  the  sun,  starting  in 
the  spring,  mounts  higher  in  the  sky  each  day  at  noon  for  three 
months,  until  it  reaches  its  greatest  elevation  at  the  summer  sol- 
stice, and  then  descends  towards  the  south,  reaching  in  the  autumn 
the  same  noonday  elevation  it  had  in  the  spring.  It  keeps  on  its 
southward  course  to  a  winter  solstice  in  December,  and  then  returns 
to  its  original  height  at  the  end  of  a  year,  marking  and  causing  the 
seasons  by  its  course.  A  year,  the  interval  between  the  successive 
returns  of  the  sun  to  the  same  position,  was  very  early  found  to 
consist  of  a  little  more  than  three  hundred  and  sixty  days. 

Nor  is  this  all.  The  sun's  motion  is  not  merely  a  north-and-south 
motion,  but  it  also  moves  eastward 1  among  the  stars  ;  for  in  the  spring 
the  stars  which  are  rising  in  the  eastern  horizon  at  sunset  are  differ- 
ent from  those  which  are  found  there  in  the  summer  or  winter.  In 
the  spring,  the  most  conspicuous  of  the  eastern  constellations  at 
sunset  are  Leo  and  Bootes  ;  a  little  later,  Virgo  appears  ;  in  the 
summer,  Ophiuchus  and  Libra  ;  still  later,  Scorpio ;  and  in  mid- 
winter, Orion  and  Taurus  are  in  the  eastern  sky  at  evening. 

173.  So  far  as  mere  appearances  go,  everything  would  be  ex- 
plained by  assuming  that  the  earth  is  at  rest  and  the  sun  moving 
around  it ;  but  equally  by  the  converse  supposition,  —  for  if  the  earth 
as  seen  from  the  sun  appears  at  any  point  in  the  heavens,  the  sun  as 
seen  from  the  earth  must  appear  in  exactly  the  opposite  point,  and 


1  Of  course  these  two  motions  of  the  sun  are  not  independent,  but  only  "com- 
ponents" of  its  motion  in  the  ecliptic  (Art.  175). 


126  THE    ECLIPTIC    AND    ZODIAC. 

must  keep  opposite,  moving  through  the  same  path  in  the  sky  (but 
six  months  behind),  and  always  in  the  same  "  angular  direction,"  if 
we  may  use  the  expression.  (Just  as  two  opposite  teeth  on  a  gear- 
wheel move  in  the  same  angular  direction,  though  at  any  moment 
they  are  moving  in  opposite  linear  directions.) 

174.  That  it  is  really  the  earth  which  moves,  and  not  the  sun,  is 
absolutely  demonstrated  by  three  phenomena  too  minute  and  delicate 
for  pre-telescopic  observations,  but  accessible  to  modern  methods. 
One  of  them  is  the  aberration  of  light;  a  second,  the  regular  annual 
backward  and  forward  shift  of  the  lines  in  star-spectra;  the  third,  the 
annual  parallax  of  the  fixed  stars.     These  can  be  explained  only  by 
the  actual  motion  of  the  earth, 

175.  The  Ecliptic.  —  By  observing  with  a  meridian  circle  daily 
the  declination  of  the  sun,  and  the  difference  between  its  right  as- 
cension and  that  of  some  star  (Flamsteed  used  a  Aquilse  for  the  pur- 
pose), we  shall  obtain  a  series  of  positions  of  the  sun's  centre  which 
can  be  plotted  on  a  celestial  globe  ;  and  we  can  thus  make  out  the 
path  of  the  sun  among  the  stars,  and  find  the  place  where  it  cuts 
the  celestial  equator,  and  the  angle  it  makes.     This  path  turns  out 
to  be  a  great  circle,  as  is  shown  by  its  cutting  the  equator  at  two 
points  just  180°  apart  (the  so-called  equinoctial  points  or  equinoxes), 
and  makes  an  angle  with  it  of  approximately  23£°.     This  great  circle 
is  called  the  ECLIPTIC,  because,  as  was  early  discovered,  eclipses 
happen  only  when  the  moon  is  crossing  it.     It  may  be  defined  as 
the  trace  of  the  plane  of  the  earth's  orbit  upon  the  celestial  sphere,  i.e., 
the  great  circle  formed  by  the  intersection  of  the  infinitely  extended 
plane  of  the  earth's  orbit  with  the  celestial  sphere. 

176.  Definitions.  —  The  angle  which  the  ecliptic  makes  with  the 
equator  is  called  the  Obliquity  of  the  ecliptic,  and  the  points  midway 
between  the  equinoxes  are  called  the  Solstices  (sol-stitium),  because 
at  these  points  the  sun  "  stands,"  or  stops  moving  in  declination  for 
a  short  time. 

Two  circles  parallel  to  the  equator,  drawn  through  the  solstices, 
are  called  the  Tropics  (Greek  TpeVw),  or  "turning-lines,"  because 
there  the  sun  turns  from  its  northward  motion  to  a  southward,  or 
vice  versa.  The  obliquity  is,  of  course,  simply  equal  to  the  surfs 
maximum  declination,  or  greatest  distance  from  the  equator,  which 
is  reached  in  June  and  December. 


CELESTIAL   LATITUDE   AND    LONGITUDE.  127 

The  ancients  were  accustomed  to  determine  it  by  means  of  the  gnomon  * 
(Art.  107).  The  length  of  the  shadow  at  noon  on  the  solstitial  days  deter- 
mines the  zenith  distance  of  the  sun  on  those  days,  and  the  difference  of  the 
zenith  distances  at  the  two  solstices  is  twice  the  angle  desired.  The  gnomon 
also  determined  for  the  ancients  the  length  of  the  year,  it  being  only  neces- 
sary to  observe  the  interval  between  days  in  the  spring  or  autumn,  when  the 
shadow  had  the  same  length  at  noon. 

177.  The  Zodiac  and  its  Signs.  —A  belt  16°  wide,  8°  on  each  side 
of  the  ecliptic,  is  called  the  Zodiac.     The  name  is  said  to  be  derived 
from  £oW,  a  living  creature,  because  the  constellations  in  it  (except 
Libra)  are  all  figures  of  animals.     It  was  taken  of  that  particular 
width  by  the  ancients  simply  because  the  moon  and  the  then  known 
planets  never  go  further  than  8°  from  the  ecliptic. 

This  belt  is  divided  into  the  so-called  SIGNS,  each  30°  in  length,  having 
the  following  names  and  symbols :  — 

( Aries,  °y  t  Libra,  A 

Spring  -3  Taurus,  8  Autumn  -3  Scorpio,  TH, 

1  Gemini,  n  C  Sagittarius,     / 

f  Cancer,  25  s  Capricornus,  V? 

Summer  -5  Leo,  St  Winter  )  Aquarius  £J 

(Virgo,     TTJJ  C  Pisces,            X 

The  symbols  are  for  the  most  part  conventionalized  pictures  of  the  ob- 
jects. The  symbol  for  Aquarius  is  the  Egyptian  character  for  water.  The 
origin  of  the  signs  for  Leo,  Virgo,  and  Capricornus  is  not  quite  clear.  It 
has  been  suggested  that  SI  is  simply  a  "  cursive  "  form  for  A,  the  initial  of 
AeW;  Wfl.  for  Ha/o  (IIap0eVos),  and  V?  for  Tp  (Tpayos). 

V  CELESTIAL   LATITUDE   AND   LONGITUDE. 

178.  Since  the    moon  and  all  the  principal  planets  always  keep 
within  the  zodiac,  the  ecliptic  is  a  very  convenient  circle  of  reference, 
and  was  used  as  such  by  the  ancients.     Indeed,  until  the  invention 
of  pendulum  clocks,  it  was  on  the  whole  more  convenient  than  the 
equator,  and  more  used. 

The  two  points  in  the  heavens  90°  distant  from  the  ecliptic  are  called 
the  Poles  of  the  ecliptic.  The  northern  one  is  in  the  constellation 

1  The  Chinese  claim  to  have  made  an  observation  of  this  kind  about  1100  B.C., 
and  the  result  given  is  very  nearly  what  it  should  have  been  at  that  time.  (The 
obliquity  changes  slightly  in  centuries.)  If  their  observation  is  genuine,  it  is 
probably  the  oldest  of  all  astronomical  records. 


128 


CELESTIAL   LATITUDE  AND   LONGITUDE. 


of  Draco,  about  half-way  between  the  stars  S  and  £  Draconis.  Now, 
suppose  a  set  of  great  circles  drawn,  like  meridians,  through  these 
poles  of  the  ecliptic,  and  hence  perpendicular  to  that  circle  ;  these 
are  Circles  of  latitude  or  secondaries  to  the  ecliptic.  The  LONGITUDE 
of  a  star  or  any  other  heavenly  body  is,  then,  the  angle  made  at  the 
pole  of  the  ecliptic,  between  the  circle  of  latitude,  which  passes  through 
the  vernal  equinox,  and  the  circle  of  latitude  passing  through  the  body; 
or,  what  comes  to  the  same  thing,  it  is  the  arc  of  the  ecliptic  included 
between  the  vernal  equinox  and  the  foot  of  the  circle  of  latitude  pass- 
ing through  the  body.  Celestial  longitude  is  always  reckoned  east- 
ward from  the  vernal  equinox,  completely  around  the  ecliptic,  so  that 
the  longitude  of  the  sun  when  10°  ivest  of  the  vernal  equinox  would  be 
written  as  350°,  and  not  as  —10°. 

The  LATITUDE  of  a  star  is  simply  its  distance  north  or  south  of 
the  ecliptic  measured  on  the  star's  circle  of  latitude. 


179. 


It  will  be  seen  that  longitude  differs  from  right  ascension  in 
being  reckoned  on  the  ecliptic  instead  of 
on  the  equator,  nor  can  it  be  reckoned 
in  time,  but  only  in  degrees,  minutes, 
and  seconds.  Latitude  differs  from  de- 
clination in  that  it  is  reckoned  from  the 
ecliptic  instead  of  from  the  equator. 

The  relation  between  right  ascension 
and  declination  on  the  one  hand,  and 
longitude  and  latitude  on  the  other, 
may  be  made  clearer  by  the  accom- 
panying diagram  (Fi£.  57),  in  which 

EG  is  the  ecliptio  and  *«  the  ei°a- 

tor,  E  being   the  vernal   equinox.      S 
being  a  star,  its  right  ascension  (a)  is 

ER  and  its  declination  (8)  is  SR  ;  its  longitude  (X)  is  EL,  and  its  lati- 
tude (/?)  is  SL.  P  and  K  are  the  poles  of  the  equator  and  ecliptic 
respectively,  and  the  circle  KPCQ,  is  the  Solstitial  Colure,  so  called. 

The  student  can  hardly  take  too  great  care  to  avoid  confusion  of  celestial 
latitude  and  longitude  with  right  ascension  and  declination  or  with  terres- 
trial latitude  and  longitude.  It  is,  of  course,  unfortunate  that  latitude  in 
the  sky  should  not  be  analogous  to  latitude  upon  the  earth,  or  celestial  longi- 
tude to  terrestrial.  The  terms  right  ascension  and  declination  are,  however, 
of  comparatively  recent  introduction,  and  found  the  ground  preoccupied, 
celestial  latitude  and  longitude  being  much  older. 


Declination. 


CELESTIAL    LATITUDE    AND    LONGITUDE. 


129 


180.  Conversion  of  X  and  ft  into  a  and  S,  or  Vice  Versa.  —  Right 
ascension  and  declination  can,  of  course,  always  be  converted  into  longitude 
and  latitude  by  a  trigonometrical  calculation.     We  proceed  as  follows:  In 
the  triangle  ERS,  right-angled  at  R,  we  have  given  ER  and  RS  (a  and  8), 
from  which  we  find  the  hypothemise  ES  and  the  angle  RES.    Next  in  the 
triangle  ELS,  right-angled  at  L,  we  have  the  hypothenuse  ES  and  the  angle 
LES,  which  is  equal  to  RES-LEQ  (LEQ  being  o>,  the  obliquity  of  the 
ecliptic).     Hence  we  easily  find  EL  and  LS. 

181.  The  Earth's  Orbit  in  Space.  —  The  ecliptic  is  not  the  earth's 
orbit,  and  must  not  be  confounded  with  it.    It  is  a  great  circle  of  the 
infinite  celestial  sphere,  the  trace  made  upon  the  sphere  by  the  plane 
of  the  earth's  orbit,  as  was  state'd  in  its  definition.     The  fact  that 
it  is  a  great  circle  gives  us  no  information  about  the  earth's  orbit, 
except  that  the  orbit  all  lies  in  one  plane  passing  through  the  sun. 
It  tells  us  nothing  as  to  its  real  form  and  size. 

By  reducing  the  observations  of  the  sun's  right  ascension  and 
declination  through  the  year  to  longitude  and  latitude  (the  latitude 
will  always  be  zero,  of  course,  except  for  some  slight  perturbations) 
and  combining  them  with  observations  of  the  sun's  apparent  diameter, 
we  can,  however,  ascertain  the  real  form  of  the  earth's  orbit  and  the 
law  of  its  motion  in  this  orbit.  But  the  size  of  the  orbit — the  scale 
of  miles — cannot  be  fixed  until  we  can  find  the  sun's  distance. 


182.  To  find  the  Form  of  the  Orbit,  we  may  proceed  thus  :  Take 
a  point  S  for  the  sun  and  draw 
from  it  a  line  80,  Fig.  58, 
directed  towards  the  vernal 
equinox  as  the  origin  of  longi- 
tudes. Lay  off  from  8  indefi- 
nite lines,  making  angles  with 
SO  equal  to  the  earth's *  longi- 
tude on  each  of  the  days  ob- 
served through  the  year  ;  i.e., 
the  angle  OS  10  is  the  longi- 
tude at  the  time  of  the  10th 
observation ;  and  so  on.  We 
shall  thus  get  a  sort  of  "  spi-  FlG  58 

der,"  Showing  the  directions  as    Determination  of  the  Form  of  the  Earth's  Orbit. 

seen  from  the  sun  on  those  days. 

Next,  as  to  relative  distances.   While  the  apparent  diameter  of  the 
sun  does  not  tell  us  its  real  distance  from  the  earth,  unless  we  first 


1  The  earth's  longitude  is  the  observed  longitude  of  the  sun  +  180°. 


130 


CELESTIAL   LATITUDE   AND   LONGITUDE. 


know  the  sun's  real  diameter  in  miles,  the  changes  in  the  apparent 
diameter  do  inform  us  as  to  the  relative  distance  of  the  earth  at 
different  times,  since  the  nearer  we  are,  the  larger  the  sun  appears,  • — 
the  distance  being  inversely  proportional  to  the  apparent  diameter 
(Art.  6).  If,  then,  we  lay  off  on  the  arms  of  our  "spider"  dis- 
tances inversely  proportional1  to  the  number  of  seconds  of  arc  in  the 
sun's  measured  diameter  at  each  date,  these  distances  will  be  pro- 
portional  to  the  true  distance  of  the  earth  from  the  sun,  and  the  curve 
joining  the  points  thus  obtained  will  be  a  true  map  of  the  earth's 

orbit,  though  without  any  scale  of 
miles  upon  it. 

'When  the  operation  is  performed, 
we  find  that  the  orbit  is  an  ellipse 
of  small  eccentricity  (about  one- 
sixtieth),  with  the  sun,  not  in  the 
centre,  but  at  one  focus.2 

183.      For  the  benefit   of    any  who 
FIG. 59. -The Ellipse.  may  not  have   studied   conic    sections 

we  define  the  ellipse.     It  is  a  curve  such 

that  the  sum  of  the  two  distances  from  any  point  on  its  circumference  to 
two  points  within,  called  the  foci,  is  always  constant,  and  equal  to  what  is 
called  the  major-axis  of  the  ellipse.  SP  +  PF=AAf,  in  Fig.  59.  AC  is 
called  the  eemi-rnajor-axis,  and  is  usually  denoted  by  A  or  a.  BC  is  the 
semi-minor-axis,  denoted  by  B  or  b.  The  eccentricity,  denoted  by  e,  is  the 

fraction  . 

AC  


Since  BS  is  equal  to  A,  SC  = 


B2  ;    and  e  = 


The  points  where  the  earth  is  nearest  to  and  most  remote  from  the 
sun  are  called  respectively  perihelion  and  aphelion,  and  the  line  that 
joins  them  is,  of  course,  the  major-axis  of  the  orbit.  This  line,  con- 
sidered as  indefinitely  produced  in  both  directions,  is  called  the  line 
of  apsides,  —  the  major-axis  being  a  limited  piece  or  "sect"  of  the 
line  of  apsides. 


184.  The  variations  of  the  sun's  diameter  are  too  small  to  be  detected 
without  a  telescope  (amounting  only  to  about  three  per  cent),  so  that 
the  ancients  were  unable  to  perceive  them.  Hipparchus,  however,  about 

1  The  distances  to  be  laid  off  are  found  by  dividing  some  arbitrarily  chosen 
constant  (say  10000")  by  the  number  of  seconds  of  arc  in  each  measured  diameter 
of  the  sun.  2  See  note  on  page  154. 


LAW   OF   THE   EARTH'S   MOTION.  131 

150  B.C.,  discovered  that  the  earth  is  not  in  the  centre  of  the  circular  orbit 
which  he  supposed  the  sun  to  describe  around  it.  Everybody  assumed,  on 
a  priori  grounds,  never  disputed  until  the  time  of  Kepler,  that  the  sun's  orbit 
must  be  a  circle  and  described  with  a  uniform  motion,  because  a  circle  is 
the  only  "  perfect  "  curve,  and  uniform  motion  the  only  perfect  motion. 
Obviously,  however,  the  sun's  apparent  motion  is  not  uniform,  because  it 
takes  186  days  for  the  sun  to  pass  from  the  vernal  equinox  to  the  autumnal 
through  the  summer  months,  and  only  179  days  to  return  during  the  win- 
ter. Hipparchus  explained  this  difference  by  the  hypothesis  that  the  earth 
is  out  of  the  centre  of  the  sun's  path. 

185.  To  find  the  Eccentricity  of  the  Orbit.  —  Having  the  greatest 
and  least  apparent  diameters  of  the  sun,  the  eccentricity,  e,  is  easily 
found.  In  Fig.  59,  since,  by  definition,  e  =  CS  -f-  (7-4,  we  have  OS  = 
CA  x  e,  or  Ae.  The  perihelion  distance  AS  is  therefore  equal  to 
A  X  (1  —  e),  and  the  aphelion  distance  SA'  to  A  (1  -f-  e).  Suppose 
now  that  the  greatest  and  least  measured  diameters  of  the  sun  are  p 
and  q.  This  gives  us  the  proportion  p:q=  A  (1-fe)  :  A  (1—  e), 
since  the  diameters  are  inversely  proportional  to  the  distances.  From 
thisweget  _  - 


The  actual  values  of  p  and  q  are  32'  36".4  and  31f  31".8,  which  give 
e  =  0.01678  :  this  is  about  g-1^,  as  has  been  stated. 

186.  To  find  the  Law  of  the  Earth's  Motion.  —  By  comparing  the 
meas  ired  apparent  diameter  with  the  differences  of  longitude  from  day 
to  day,  we  can  also  deduce  the  law  of  the  earth's  motion.    On  making 
a  table  of  daily  motions  and  apparent  diameters,  we  find  that  these 
daily  motions  vary  directly  as  the  squares  of  the  diameters;  from  which 
it  directly  follows  that  the  earth  moves 

in  such  a  way  that  its  radius-vector 
describes  areas  proportional  to  the  times  f/. 
(a  law  which  Kepler  first  brought  to 
light  in  1609).  The  radius-vector  is 
the  line  which  joins  the  earth  to  the 
sun  at  any  moment. 

187.  Consider   a   small   elliptical  sec- 
tor,  dSc  (Fig.  60),  described  by  the  earth 

•4.     t  I-  -D  j-         i  Equable  Description  of  Areas. 

in  a  unit  of  time.     Regarding  it  as  a  trian- 

gle, its   area  is  given  by  the  formula  \  Sc  X  Sd  sin  cSd  ;   and  calling  this 

angle  6  (which  will  be  very  small),  and  considering  that  in  so  short  a  time 


132  KEPLER'S  PROBLEM. 

Sd  and  Sc  would  remain  sensibly  equal,  each  being  equal  to  R  (the  radius- 
vector  at  the  middle  point  of  the  arc),  this  formula  becomes, 

Area  of  sector  =±  R20. 
Now,  calling  the  sun's  apparent  diameter  Z>,  we  have 

1  k 

R  varies  as  —  or  R  =  — • , 

(k  being  a  constant,  and  depending  on  the  sun's  diameter  in  miles) ; 

k2 
whence  R2=——. 

But  our  measurements  show  that  6  =  k^D2,   k^  being  another  constant. 
Substitute  these  values  of  R2  and  6  in  the  formula  for  the  area,  and  we  have 

Area  of  sector  =  |  —^  x  ^  D2  =  £  k*klt 

a  constant ;  that  is,  the  area  described  by  the  radius-vector  in  a  unit  of  time 
is  always  the  same.  The  planet  near  perihelion  moves  so  much  faster,  that 
the  areas  aSb,  cSd,  and  eSf  are  all  equal  to  each  other,  if  the  arcs  are  de- 
scribed in  the  same  time. 

188.  Kepler's  Problem. — As  the  case  stands  so  far,  this  is  a  mere 
fact  of  observation  ;  but  as  we  shall  see  hereafter,  and  as  was  demon- 
strated by  Newton,  the  fact  shows 
that  the  earth  moves  under  the  ac- 
tion of  a  force  always  directed  in  line 
with  the  sun.  In  such  a  case  the 
"equable  description  of  areas"  is  a 
necessary  mechanical  consequence. 
It  is  true  in  every  case  of  elliptical 
motion,  and  enables  us  to  find  the 
FIG.  ei. -Kepler's  Problem.  position  of  the  earth  or  any  planet 

in   its   orbit   at   any   time,  when  we 

once  know  the  time  of  its  orbital  revolution  (technically  the  period) , 
and  the  time  when  it  was  at  perihelion.  Thus,  the  angle  ASP  (Fig.  61), 
which  is  called  the  Anomaly  of  the  planet,  must  be  such  that  the  area  of 
the  elliptical  sector  ASP  will  be  that  portion  of  the  whole  ellipse  which 

is  represented  by  the  fraction  — ,  t  being  the  number  of  days  since  the 

planet  last  passed  the  perihelion,  and  T  the  number  of  days  in  the 
whole  period.  For  instance,  if  the  earth  last  passed  perihelion  on 
Dec.  31  (which  it  did),  its  place  on  May  1  must  be  such  that  the 


ANOMALY   AND   EQUATION   OF    THE   CENTKE.  133 


sector  ASP  will  be  ^B^}  of  the  whole  of  the  earth's  orbit  ;  since  from 
December  31  to  May  1  is  121  days.  The  solution  of  this  problem, 
known  as  "Kepler's  problem"  leads  to  transcendental  equations,  and 
lies  beyond  our  scope. 

See  Watson's  "  Theoretical  Astronomy,"  pp.  53  and  54,  or  any  other  simi- 
lar work  ;  also  Appendix,  Arts.  1002  and  1003. 

189.  Anomaly  and  Equation  of  the  Centre.  —  The  angle  ASP, 
which  has  been  termed  simply  the  "Anomaly"  is  strictly  the  true 
Anomaly,  as  distinguished  from  the  mean  Anomaly.  The  former 
may  be  denned  as  the  angle  actually  made  at  any  time  by  the  radius- 
vector  of  a  planet  with  the  line  of  apsides,  the  angle  being  reckoned 
from  the  perihelion  point  completely  around  in  the  direction  of  the 
planet's  motion.  The  mean  Anomaly  is  what  the  Anomaly  would  be 
at  the  given  moment  if  the  planet  had  moved  with  uniform  angular 
velocity,  completing  the  orbit  in  the  same  period,  and  passing  perihelion 
at  the  same  time,  as  it  actually  does.  The  difference  between  the 
two  anomalies  is  called  the  Equation  of  the  Centre.  This  is  zero  at 
perihelion  and  aphelion,  and  a  maximum  midway  between  them. 
In  the  case  of  the  sun,  its  greatest  value  is  nearly  2°,  the  sun  getting 
alternately  that  amount  ahead  of,  and  behind,  the  position  it  would 
occupy  if  its  apparent  daily  motion  were  uniform. 


THE  SEASONS. 

190.  The  Seasons.  —  The  earth  in  its  motion  around  the  sun 
always  keeps  its  axis  parallel  to  itself,  for  the  mechanical  reason 
that  a  revolving  body  necessarily  maintains  the  direction  of  its  axis 
invariable,  unless  disturbed  by  extraneous  force,  as  is  very  prettily 
illustrated  by  the  gyroscope.  Fig.  61*  shows  the  way  in  which  the 
north  pole  of  the  earth  is  inclined  with  reference  to  the  sun  at  dif- 
ferent seasons  of  the  year. 

About  March  20  the  earth  is  so  situated  that  the  plane  of  its 
equator  passes  through  the  sun,  the  sun's  declination  being  zero  on 
that  day.  At  that  time,  the  line  which  separates  the  illuminated 
portions  of  the  earth  passes  through  the  two  poles  (as  shown  in 
Fig.  63  -S),  and  day  and  night  are  everywhere  equal.  The  same  is 
again  true  of  the  22d  of  September,  when  the  sun  is  at  the  autumnal 
equinox  on  the  opposite  side  of  the  orbit. 


134 


THE    SEASONS. 


About  the  21st  of  June  the  earth  is  so  situated  that  its 
north  pole  is  inclined  towards  the  sun  by  about  23-J-0,  which 
is  the  sun's  northern  declination  on  that  date.  As  shown  in 
Fig.  62  A,  the  south  pole  is  then  in  the  obscure  half  of  the 
earth's  globe. 


lutumnal  Equinox 


Vernal  Equinox 

FIG.  61*.  —  The  Seasons. 


The  north  pole  then  receives  sunlight  all  day  long  ;  and  in  all 
portions  of  the  northern  hemisphere  the  day  is  longer  than  the 
night,  the  difference  between  the  day  and  night  depending  upon 
the  latitude  of  the  place,  while  in  the  southern  hemisphere  the 


FIG.  62.  —  Position  of  Pole  at  Solstice  and  Equinox. 


DIUKNAL  PHENOMENA  NEAR  THE  POLE.  135 

days  are  shorter  than  the  nights.  At  the  time  of  the  winter  solstice 
these  conditions  are  reversed.  At  the  equator  (of  the  earth)  the  day 
and  night  are  equal  at  all  times  of  the  year.  The  sun  when  in  north- 
ern declination  of  course  always  rises  at  a  point  on  the  horizon  north 
of  east,  and  sets  at  a  point  north  of  west,  so  that  for  a  portion  of 
the  time  each  day  it  shines  on  the  north  side  of  a  house. 

191.  Diurnal  Phenomena  near  the  Pole.  —  At  the  north  pole,  where 
the  celestial  pole  is  in  the  zenith,  and  the  diurnal  circles  are  parallel  with 
the  horizon,  the  sun  will  maintain  the  same  elevation  all  day  long,  except 
for  the  slight  change  caused  by  the  variation  of  its  declination  in  twenty- 
four  hours.     The  sun  will  appear  on  the  horizon  at  the  date  of  the  vernal 
equinox  (in  fact,  about  three  days  before,  on  account  of  refraction),  and 
slowly  wind  upward  in  the  sky  until  it  reaches  its  maximum  elevation  of 
23|°  on  June  21.     Then  it  will  retrace  its  course  until  a  day  or  two  after 
the  autumnal  equinox,  when  it  sinks  out  of  sight. 

At  points  between  the  north  pole  and  the  polar  circle  the  sun  will  appear 
above  the  horizon  earlier  in  the  year  than  March  20,  and  will  rise  and  set 
daily  until  its  declination  becomes  equal  to  the  observer's  distance  from  the 
pole,  when  it  will  make  a  complete  circuit  of  the  heavens,  touching  the  hori- 
zon at  midnight  at  the  northern  point ;  and  after  that  never  setting  again 
until  it  reaches  the  same  declination  in  its  southward  course  after  passing 
the  solstice.  From  that  time  it  will  again  rise  and  set  daily  until  it  reaches 
a  southern  declination  just  equal  to  the  observer's  polar  distance,  when  the 
long  night  begins ;  to  continue  until  the  sun,  having  passed  the  southern 
solstice,  returns  again  to  the  same  declination  at  which  it  made  its  appear- 
ance in  the  spring.  At  the  polar  circle  itself  (or,  more  strictly  speaking, 
owing  to  refraction,  about  one-half  a  degree  south  of  it)  the  "midnight  sun'* 
will  be  seen  on  just  one  day  in  the  year,  the  day  of  the  summer  solstice ; 
and  there  will  also  be  one  absolutely  sunless  day,  viz.,  the  day  of  the  winter 
solstice.  The  same  remarks  apply  in  the  southern  hemisphere,  by  making 
the  obvious  changes. 

192.  Effects  on  Temperature,  —  The  changes  in  the  duration  of 
"insolation"  (exposure  to  sunshine)   at  any  place  involve  changes 
of  temperature  and  other  climatic  conditions,  thus  producing  the  sea- 
sons.   Taking  as  a  standard  the  amount  of  heat  received  in  twenty-four 
hours  on  the  day  of  the  equinox,  it  is  clear  that  the  surface  of  the 
soil  at  any  place  in  the  northern  hemisphere  will  receive  more  than 
this  average  amount  of  heat  whenever  the  sun  is  north  of  the  celestial 
equator,  for  two  reasons. 

1.  Sunshine  lasts  more  than  half  the  day. 

2,  The  mean  elevation  of  the  sun  during  the  day  is  greater  than 


136 


TIME   OF   HIGHEST   TEMPERATURE. 


FIG.  62*. 

Effect  of  Sun's  Elevation  on  Amount 
of  Heat  Imparted  to  the  Soil. 


when  it  is  at  the  equinoxes,  since  it  is  higher  at  noon,  and  in  any 
case  reaches  the  horizon  at  rising  and  setting.  Now,  the  more 
obliquely  the  rays  strike,  the  less  heat  they  bring  to  each  square 

inch  of  surface.  A  beam  of  sunshine 
of  given  cross-section,  ABCD,  is 
spread  over  a  larger  area  when  it 
strikes  obliquely  than  when  it  is  ver- 
tical, as  is  obvious  from  Fig.  62*,  and 
its  heating  efficiency  is  correspond- 
ingly diminished.  If  Q  is  the  amount 
of  heat  per  square  inch  brought  by 
the  ray  when  falling  perpendicularly, 
as  on  the  surface  A  C,  then  on  Ac  (on 
which  it  strikes  at  the  angle  h,  equal 
to  the  sun's  altitude)  the  amount  per  square  inch  will  be  Q  X  sin  h, 
since  AB  =  Ab  X  sin  h.  This  difference  in  favor  of  the  more  nearly 
vertical  rays  is  exaggerated  by  the  absorption  of  heat  in  the  atmos- 
phere, because  rays  that  are  nearly  horizontal  have  to  traverse  a 
much  greater  thickness  of  air  before  reaching  the  ground. 

For  these  two  reasons,  at  a  place  in  the  northern  hemisphere,  the 
temperature  rises  rapidly  as  the  sun  comes  north  of  the  equator,  thus 
giving  us  our  summer. 

193,  Time  of  Highest  Temperature.  —  We,  of  course,  receive  the 
most  heat  per  diem  at  the  time  of  the  summer  solstice  ;  but  this  is 
not  the  hottest  time  of  the  summer,  for  the  obvious  reason  that  the 
weather  is  then  all  the  time  getting  hotter,  and  the  maximum  will 
not  be  reached  until  the  increase  ceases;  that  is,  not  until  the  amount 
of  heat  lost  in  the  night  equals  that  stored  up  by  day. 

If  the  earth's  surface  threw  off  the  same  amount  of  heat  hourly  whether 
it  were  hot  or  cold,  then  this  maximum  would  not  come  until  the  autumnal 
equinox.  This,  however,  is  not  the  case.  The  soil  loses  heat  faster  when 
warm  than  it  does  when  cold,  the  loss  being  nearly  proportional  to  the  dif- 
ference between  the  temperature  of  the  soil  and  that  of  surrounding  space 
(Newton's  law  of  cooling)  ;  and  so  the  time  of  the  maximum  is  made  to 
come  not  far  from  the  end  of  July,  or  the  first  of  August,  in  our  latitude. 
For  similar  reasons  the  minimum  temperature  of  winter  occurs  about  Feb- 
ruary 1,  about  half-way  between  the  solstice  and  the  vernal  equinox.  Since, 
however,  our  weather  is  not  entirely  "  made  on  the  spot  where  it  is  used," 
but  is  affected  by  winds  and  currents  that  come  from  great  distances,  the 
actual  time  of  the  maximum  temperature  cannot  be  determined  by  any  mere 
astronomical  considerations,  but  varies  considerably  from  year  to  year. 


1 t- 
DIFFERENCE   BETWEEN   SEASONS.  137 

194.  Difference  between  Seasons  in  Northern  and  Southern  Hemi- 
spheres. —  Since  in  December  the  distance  of  the  earth  from  the  sun 
is  about  three  per  cent  less  than  it  is  in  June,  the  earth  (as  a  whole) 
receives  hourlj'  about  six  per  cent  more  heat  in  December  than  in 
June,  the  heat  varying  inversely  as  the  square  of  the  distance.     For 
this  reason  the  southern  summer,   which  occurs  in  December  and 
January,  is   hotter  than    the  northern.      It  is,  however,  seven  days 
shorter,  because  the  earth  moves  more  rapidly  in  that  part   of  its 
orbit.     The  total  amount  of  heat  per  acre,  therefore,  received  during 
the  summer  is  sensibly  the  same  in  each  hemisphere,  the  shortness 
of  the  southern  summer  making  up  for  its  increased  warmth. 

195.  The  southern  winter,  however,  is  both  longer  and  colder  than  the 
northern ;    and  it  has  been  vigorously  maintained  by  certain   geologists, 
that,  on  the  whole,  the  mean  annual  temperature  of  the  hemisphere  which 
has  its  winter  at  the  time  when  the  earth  is  in  aphelion  is  lower  than  that 
of  the  opposite  one.    It  has  been  attempted  to  account  for  the  glacial  epochs 
in  this  way.     It  is  certain  that  at  present,  at  any  place  in  the  southern  hem- 
isphere, the  difference  between  the  maximum  temperature  of  summer  and 
the  minimum  of  winter  must  be  greater  than  in  the  case  of  a  station  in  the 
northern  hemisphere,  similarly  situated  as  to  elevation,  etc.     We  say  "  at 
present "  because,  on  account  of  certain  slow  changes  in  the  earth's  orbit,  to 
be  spoken  of  immediately,  the  state  of  things  will  be  reversed  in  about  ten 
thousand  years,  the  northern  summer  being  then  the  hotter  and  shorter  one. 

196.  Secular  Changes  in  the  Orbit  of  the  Earth.  —  The  orbit  of 
the  earth  is  not  absolutely  unchangeable  in  form  or  position,  though 
it  is  so  in  the  long  run  as  regards  the  length  of  its  major  axis  and  the 
duration  of  the  year. 

197.  1.  Change  in  Obliquity  of  the  Ecliptic.  — The  ecliptic  slightly 
and  very  slowly  shifts  its  position   among   the  stars,  thus  altering 
the  latitudes   of   the  stars  and  the  angle  between   the   ecliptic  and 
equator,    i.e.,  the  obliquity   of   the   ecliptic.      This  obliquity  is  at 
present  about  24f  less   than   it   was   2000   years  ago,  and   is   still 
decreasing  about  half  a  second  a  year.     It  is  computed   that   this 
diminution  will  continue. for  about  15.000  years,  reducing  the  obli- 
quity to  22i°,  when  it  will  begin  to  increase.     The  whole  change, 
according  to  J.  Herschel,  can  never  exceed  about  1°  20'  on  each  side 
of  the  mean. 

198.  2.   Change  of  Eccentricity.  —  At  present  the  eccentricity  of 
the  earth's  orbit  (which  is  now  0.0168)  is  also   slowly  diminishing. 


138  EQUATION   OF  TIME. 

According  to  Leverrier,  it  will  continue  to  decrease  for  about  24,000 
years,  until  it  becomes  0.003,  and  the  orbit  will  be  almost  circu- 
lar.' Then  it  will  increase  again  for  40,000  years,  until  it  becomes 
0.02. 

In  this  way  the  eccentricity  will  oscillate  backwards  and  forwards,  always, 
however,  remaining  between  zero  and  0.07;  but  the  oscillations  are  not 
equal  either  in  amount  or  time,  and  so  cannot  properly  be  compared  to  the 
"vibrations  of  a  mighty  pendulum,"  which  is  rather  a  favorite  figure  of 
speech. 

199.  3.  Revolution  of  the  Apsides  of  the  Earth's  Orbit.  — The  line 
of  apsides  of  the  orbit,  which  now  stretches  in  both  directions  towards 
the   constellations   of   Sagittarius   and   Gemini,  is  also  slowly  and 
steadily  moving  eastward,  and  at  a  rate  wh;ch,  if  it  were  constant, 
would  carry  it  around  the  circle  in  about  108,000  years. 

200.  These  so-called  "secular"  changes  are  due  to  the  action  of 
the  other  planets  upon  the  earth.    Were  it  not  for  their  attraction,  the 
earth  would  keep  her  orbit  with  reference  to  the  sun  strictly  unaltered 
from  age  to  age,  except  that  possibly  in  the  course  of  millions  of 
years  the  effects  of  falling  meteoric  matter  and  the  attraction  of  the 
nearer  fixed  stars  might  make  themselves  felt. 

Besides  these  secular  perturbations  of  the  earth's  orbit,  the  earth  itself  is 
continually  being  slightly  disturbed  in  its  orbit.  On  account  of  its  con- 
nection with  the  moon,  it  oscillates  each  month  a  few  hundred  miles  above  and 
below  the  true  plane  of  the  ecliptic,  and  by  the  action  of  the  other  planets  it  is 
sometimes  set  forwards  or  backwards  to  the  extent  of  a  few  thousand  miles. 
Of  course  every  such  change  produces  a  corresponding  slight  change  in  the 
apparent  position  of  the  sun. 

201.  Equation  of  Time. — We  have  stated  a  few  pages   back 
(Art.  Ill),  that  the  interval  between  the  successive  passages  of  the 
sun   across   the   meridian  is  somewhat  variable,  and   that  for  this 
reason  apparent  solar,  or  sun-dial,  days  are  unequal.     On   this  ac- 
count mean  time  has  been  adopted,  which  is  kept  by  a  "fictitious" 
or   "mean"   sun   moving  uniformly   in   the   equator   at   the   same 
average   rate  as  that  of  the  real  sun  in  the  ecliptic.     The  hour- 
angle   of  this   mean   sun   is,    as   has   been   already    explained,   the 
local    mean    time   (or    clock    time);    while    the    hour-angle    of    the 
real  sun  is  the  apparent  or  sun-dial  time.     The  Equation  of  Time  . 
is  the  difference  between  these  two  times,  reckoned  as  plus  when 


EQUATION   OF   TIME. 


139 


the  sun-dial  is  slower  than  the  clock,  and  minus  when  it  is  faster. 
It  is  the  correction  which  must  be  added  (algebraically)  to  apparent 
time  in  order  to  get  mean  time.  As  it  is  the  difference  between  the 
two  hour-angles,  it  may  also  be  denned  as  the  difference  between  the 
right  ascensions  of  the  'mean  sun  and  the  true  sun  ;  or  as  a  formula 
we  may  write  :  E  =  at  —  ain)  in  which  am  is  the  right  ascension  of  the 
mean  sun,1  and  at  of  the  true  sun. 

The  principal  causes  of  this  difference  are  two  :  — 

202.  1.    The  Variable  Motion  of  the  Sun  in  the  Ecliptic,  due  to 
the  Eccentricity  of  the  Earth's  Orbit.  —  Near  perihelion,  which  occurs 
about  December  31,  the  earth's  orbital  motion  is  most  rapid  accord- 
ing to  the  law  of  "  equal  areas  "  (Art.  186).     This  makes  the  sun's 
apparent  eastward  motion  (in  longitude)  correspondingly  greater, 
and  hence  at  this  time  the  apparent  solar  days  exceed  the  sidereal 
by  more  than  the  average  amount,  making  the  sun-dial  days  longer 
than  the  mean.     (The  average  solar  day,  it  will  be  remembered,  is 
3m  56s  longer  than  the  sidereal.)     The  sun-dial  will  therefore  lose 
time  at  this  season,  and  will  continue  to  do  so  for  about  three 
months,  until  the  angular  motion  of  the  sun  falls  to  its  mean  value. 
Then  it  will  gain  until  aphelion,  when,  if  the  clock  and  the  sun 
were  started  together  at  perihelion,  they  will  once  more  be  together. 
During  the  next  half  of  the  year  the  action  will  be  reversed.     Thus, 
twice  a  year,  so  far  as  -the  eccentricity  of  the  earth's  orbit  is  con- 
cerned, the  clock  and  sun  would  be  together  at  perihelion  and 
aphelion,  while  half-way  between  they  would  differ  by  about  eight 
minutes;  the  equation  of  time  (so  far  as  due  to  this  cause  only)  being 
about  +  3  minutes  in  the  spring,  and  —  8  minutes  in  the  autumn. 

203.  2.    The*  Inclination  of  the 
Ecliptic  to  the  Equator.  —  Even  if 
the  sun's  (apparent)  motion  in  lon- 
gitude (i.e.,  along  the  ecliptic)  were 
uniform,  its  motion  in  right  ascen- 
sion would  be  variable.      If   the 
true  and  fictitious  suns  started  to- 
gether at  the  equinox,  they  would 
indeed  be  together  at  the  solstices 

and  at  the  other  equinox,  because 

*  FIG.  63. 

It    IS    jUSt    180°    from    equinox    to        Effect  of  Obliquity  of  Ecliptic  in  produc- 

equinox,  and  the  solstices  are  ex-  ing  Equation  of  Time. 


1  am  always  also  equals  the  sun's  mean  longitude. 


140 


EQUATION   OF   TIME. 


actly  half-way  between  them.  But  at  intermediate  points,  between 
the  equinoxes  and  solstices,  they  would  not  be  together  on  the  same 
hour-circle.  This  is  best  seen  by  taking  a  celestial  globe  and  mark- 
ing on  the  ecliptic  a  point,  m,  half-way  between  the  vernal  equinox 
and  the  solstice,  and  also  marking  a  point,  n,  on  the  equator,  45° 
from  the  equinox.  It  will  at  once  be  seen  that  the  former  point,  m 
in  Fig.  63,1  is  west  of  n,  so  that  m  in  the  daily  westward  motion  of 
the  sky  will  come  to  the  meridian  first  ;  in  other  words,  when  the 
sun  is  half-way  between  the  vernal  equinox  and  the  summer  solstice, 
the  sun-dial  is  faster  than  the  clock,  and  the  equation  of  time  is 
minus.  The  difference,  measured  by  the  arc  m'n,  amounts  to  nearly 
ten  minutes  ;  and  of  course  the  same  thing  holds,  mutatis  mutandis, 
for  the  other  quadrants. 

204.  Combination  of  the  Effects  of  the  Two  Causes.  —  We  can  rep- 
resent graphically  these  two  components  of  the  equation  of  time 
and  the  result  of  their  combination  as  follows  (Fig.  64)  :  — 

The  central  horizontal  line  is  a  scale  of  dates  one  year  long,  the 
letters  denoting  the  beginning  of  each  month.  The  dotted  curve 


+10" 


Dd 


^ 


M 


\ 


N 


D 


FIG.  64.  —  The  Equation  of  Time. 

shows  the  equation  of  time  due  to  the  eccentricity  of  the  earth's  or- 
bit, above  considered.  Starting  at  perihelion  on  December  31,  this 
component  is  then  zero,  rising  from  there  to  a  value  of  about  -f  8m 
on  April  2,  falling  to  zero  on  June  30,  and  reaching  a  second  maxi- 
mum of  —  8m  on  October  1.  In  the  same  way  the  broken-line  curve 


1  Fig.  63  represents  a  celestial  globe  viewed  from  the  west  side,  the  axis  being 
vertical,  and  E,  the  pole  of  the  ecliptic,  on  the  meridian,  while  E  is  the  vernal 
equinox. 


EQUATION   OF   TIME.  141 

denotes  the  effect  of  the  obliquity  of  the  ecliptic,  which,  by  itself 
alone  considered,  would  produce  an  equation  of  time  having  four 
maxima  of,  approximately,  10m  each,  about  the  6th  of  February,  May, 
August,  and  November  (alternately  -j-  and  — ),  and  reducing  to  zero 
at  the  equinoxes  and  solstices.1 

The  full-lined  curve  represents  their  combined  effect,  and  is  con 
structed  by  making  its  ordinate  at  each  point  equal  to  the  sum 
(algebraic)  of  the  ordinates  of  the  two  other  curves.  At  the  1st  of 
February,  for  instance,  the  distance,  F  3,  in  the  figure  =  Fl  +  F2. 
So,  also,  M  6  =  M  4  +  M  5  ;  the  components,  however,  in  this  case 
have  opposite  signs,  so  that  the  difference  is  actually  taken. 

The  equation  of  time  is  zero  four  times  a  year,  viz.,  on  April  15, 
June  14,  September  1,  and  December  24.  The  maxima  are  Febru- 
ary 11,  +  14ra  32s;  May  14,  —  3m  55s;  July  26,  +  6ra  128,  and 
November  2,  —  16m  18s.  But  the  dates  and  amounts  vary  slightly 
from  year  to  year. 

The  two  causes  above  discussed  are  only  the  principal  ones  effective  in 
producing  the  equation  of  time,  but  all  other  causes  combined  never  alter 
the  equation  by  more  than  a  few  seconds. 

205.  Precession  of  the  Equinoxes.  —  The  length  of  year  was 
found  in  two  ways  by  the  ancients  :  — 

1.  By  the  gnomon,  which  gives  the  time  of  the  equinox  and 

solstice  ;    and 

2.  By  observing  the  position  of  the  sun  with  reference  to  the 
stars,  —  their  rising  and  setting  at  sunrise  or  sunset. 

Comparing  the  results  of  observations  made  at  long  intervals, 
Hipparchus  (120  B.C.)  found  that  the  two  do  not  agree ;  the  former 
year  (from  equinox  to  equinox)  being  20m  23s  shorter  than  the  other 
(according  to  modern  data).  The  equinox  moves  westward  on  the 
ecliptic,  as  if  it  advanced  to  meet  the  sun  on  each  annual  return. 
He  therefore  called  its  motion  the  "precession  "  of  the  equinoxes. 

On  comparing  the  latitudes  of  the  stars  in  the  time  of  the  ancient 
astronomers  with  the  present  latitudes,  we  find  that  they  have 
changed  very  slightly  indeed;  and  we  know  therefore  that  the 
ecliptic  maintains  its  position  sensibly  unaltered.  On  the  other 
hand,  the  longitudes  of  the  stars  have  been  found  to  increase  regu- 
larly at  the  rate  of  about  50".2  annually,  —  fully  30°  in  the  last 
2000  years.  Since  longitudes  are  reckoned  from  the  equinox  (the 

1  The  fact  that  our  afternoons  begin  to  lengthen  about  December  8  is  due  to 
the  rapid  decrease  of  the  equation  of  time  then  in  progress. 


142  PRECESSION. 

intersection  between  the  ecliptic  and  equator),  and  since  the  ecliptic 
does  not  move,  it  is  evident  that  the  motion  must  be  in  the  celestial 
equator  ;  and  accordingly  we  find  that  both  the  right  ascension  and 
the  declination  of  the  stars  are  constantly  changing. 

206.  Motion  of  the  Pole  of  the  Equator  around  the  Pole  of  the 
Ecliptic.  —  The  obliquity  of  the  ecliptic,  which  equals  the  distance 
in  the  sky  between  the  pole  of  the  equator  and  the  pole  of  the 
ecliptic  (Art.  178),  has  remained  nearly  constant.     Hence  the  pole 
of  the  equator  must  be  describing  a  circle  around  the  pole  of  the 
ecliptic  in  a  period  of  about  25800  years  (360°  divided  by  50".2). 
The  pole  of  the  ecliptic  has  remained  nearly  fixed  among  the  stars, 
but  the  other  pole  has  changed  its  position  materially.     At  present 
the  pole  star  is  about  1J°  from  the  pole.     At  the  time  of  the  star 
catalogue  of  Hipparchus  it  was  12°  distant  from  it,  and  during  the 
next  two  centuries  it  will  approach  to  within  about  30',  after  which 
it  will  recede. 

207.  If  upon  a  celestial  globe  we  take  the  pole  of  the  ecliptic  as  a  cen- 
tre, and  describe  about  it  a  circle  with  a  radius  of  23£°,  we  shall  get  the 
approximate   track   of   the   celestial   pole,  and  shall  find  that   the   circle 
passes  yery  near  the  star  a  Lyrse,  which  will  be  the  pole  star  about  12000 
years  hence.     Reckoning  backwards,  we  find  that  some  4000  years  ago 
a  Draconis  was  the  pole  star ;    and  it  is  a  curious  circumstance  that  certain 
of  the  tunnels  in  the  pyramids  of  Egypt  face  exactly  to  the  north,  and  slope 
at  such  an  inclination  that  this  star  at  its  lower  culmination  would  have 
been  visible  from  their  lower  end  at  the  date  when  the  pyramids  are  sup- 
posed to  have  been  built.     It  is  probable  that  these  passages  were  arranged 
to  be  used  for  the  purpose  of  observing  the  transits  of  this  star. 

Because  of  the  changes  in  the  position  of  the  ecliptic  (Art.  197)  the  track 
of  the  pole  among  the  stars  is  not  a  perfect  circle ;  its  centre  is  not  fixed. 

208.  Effect  of  Precession  upon  the  Signs  of  the  Zodiac.  —  Another 
effect  of  precession  is  that  the  signs  of  the  zodiac  do  not  now  agree  with  the 
constellations  which  bear  the  same  name.     The  sign  of  Aries  is  now  in  the 
constellation  of  Pisces  ;   and  so  on,  each  sign  having  "backed,"  so  to  speak, 
into  the  constellation  west  of  it. 

209.  Physical  Cause  of  Precession.  —  The  physical  cause  of  this 
slow  conical  rotation  of  the  earth's  axis  around  the  pole  of  the 
ecliptic  lies  in  the  two  facts  that  the  earth  is  not  exactly  spherical, 
and  that  the  attractions  of  the  sun  and  moon1  act  upon  the  equatorial 

1  The  planets,  by  their  action  upon  the  plane  of  the  earth's  orbit  (Art.  197), 
slightly  disturb  the  equinox  in  the  opposite  direction.  This  effect  amounts  to 
about  0'M6  annually. 


PRECESSION. 


143 


ring  of  matter  which  projects  above  the  true  sphere,  tending  to  draw 
the  plane  of  the  equator  into  coincidence  with  the  plane  of  the  ecliptic 
by  their  greater  attraction  on  the  nearer  portions  of  the  ring.  The 
action  is  just  what  it  would  be  if  a  spheroidal  ball  of  iron  of  the 
shape  of  the  earth  had  a  magnet  brought  near  it.  The  magnet,  as 
illustrated  in  Fig.  65,  would  tend  to  draw  the  plane  of  the  equator 
into  the  line  CM  joining  its  pole  with  the  centre  of  the  globe,  be- 
cause it  attracts  the  nearer  portion  of  the  equatorial  protuberance 
at  E  more  strongly  than  the  remoter  at  Q.  If  it  were 
not  for  the  earth's  rotation,  this  attraction  would 
bring  the  two  planes  of  the  ecliptic  and  equator  to- 
gether ;  but  since  the  earth  is  spinning  on  its  axis,  we 
get  the  same  result  that  we  do  with  the  whirling 


FIG.  65. 

Effect  of  Attraction 
on  a  Spheroid. 


FIG.  66. 

Precession  illustrated  by  the  Gyroscope. 


wheel  of  a  gyroscope  by  hanging  a  weight  at  one  end  of  the  axis. 
We  then  have  the  result  of  the  combination  of  two  rotations  at  right 
angles  with  each  other,  one  the  whirl  of  the  wheel,  the  other  the 
"tip"  which  the  weight  tends  to  give  the  axis.  (Physics,  pp.  53-54.) 

.  210.  In  this  case,  if  the  wheel  of  the  gyroscope  is  turning  swiftly 
clock-wise,  as  seen  from  above  (Fig.  66),  the  weight  at  the  (lower) 
end  of  the  axis  will  make  the  axis  move  slowly  around,  counter-clock- 
wise, without  at  all  changing  its  inclination.  If  we  regard  the  hori- 
zontal plane  passing  through  the  gyroscope  as  representing  the 
ecliptic,  and  the  point  in  the  ceiling  vertically  above  the  gyroscope 
as  the  pole  of  the  ecliptic,  the  line  of  the  axis  of  the  wheel  produced 
upward  would  describe  on  the  ceiling  a  circle  around  this  imaginary 
ecliptic  pole,  acting  precisely  as  does  the  pole  of  the  earth's  axis  in 
the  sky.  The  swifter  the  wheel's  rotation,  the  slower  would  be  this 


144  PRECESSION. 


"precessional"  motion  of  its  axis;  and  of  course,  the  rate  of  motion 
also  depends  upon  the  magnitude  of  the  suspended  weight. 

211.  A  full  treatment  of  the  subject  would  be  too  complicated  for  our 
pages.     An  elementary  notion  of  the  way  the  action  takes  place,  correct  as 
far  as  it  goes,  is  easily  obtained  by  reference  to  Fig.  67.     Let  XY  be  the 
axis  of  the  gyroscope,  the  wheel  being  seen  in  section  edge-wise,  and  the  eye 
being  on  the  same  level  as  the  centre  of  the  wheel ;  the  wheel  turning  so 
that  the  point  B  is  coming  towards  the 

observer.  Now,  suppose  a  weight  hung 
on  the  lower  end  of  the  axis.  If  the 
wheel  were  not  turning,  the  point  B 
would  come  to  some  point  F  in  the  same 
time  it  now  takes  to  reach  C  (that  is, 
after  a  quarter  of  a  revolution).  By  com- 
bination of  the  two  motions  it  will  come 
to  a  point  K  at  the  end  of  the  same  time, 
having  crossed  the  horizontal  plane  AD  \1 

at  L  ;    and  this  can  be  effected  only  by  a  FIG.  67. 

backward  "skewing  around"  of  the  whole      Kegression  of  the  Gyroscope  Wheel, 
wheel,  axis  and  all.     This  does  not,  of 

course,  explain  why  the  inclination  of  the  axis  does  not  change  under  the 
action  of  the  weight,  but  is  only  a  very  partial  illustration,  showing  merely 
why  the  plane  of  the  wheel  regresses.  A  complete  discussion  would  require 
the  consideration  of  the  motion  of  every  point  on  the  wheel  by  a  thorough 
and  difficult  analytical  treatment. 

The  motions  of  the  earth's  axis,  discussed  in  Art.  108,  do  not  displace  the 
celestial  pole  with  reference  to  the  stars,  and  must  not  be  confounded  with 
precession. 

212.  Why  Precession  is  so  Slow.  —  The  slowness  of  the  precession 
depends  on  three  things  :    (a)  the  enormous  "moment  of  rotation" 
of  the  earth  —  a  point  on  the  equator  moves  with  the  speed  of  a 
cannon  ball ;    (b)  the  smallness  of  the  mass  (compared  with  that  of 
the  whole  earth)  of  the  protuberant  ring  to  which  precession  is  due ; 
and  (c)   the  minuteness  of  the  force  which  tends  to  bring  this  ring 
into  coincidence  with  the  ecliptic,  a  force  which  is  not  constant  and 
persistent,  like  the  weight  hung  on  the  gyroscope  axis,  but  very 
variable. 

213.  The  Equation  of  the  Equinox.  —  Whenever  the  sun  is  in  the 

plane  of  the  equator  (which  is  twice  a  year,  at  the  time  of  the  equinoxes), 
the  sun's  precessional  force  disappears  entirely,  its  attraction  then  having  no 
tendency  to  draw  the  equator  out  of  its  position.  The  moon's  action,  on 
account  of  her  proximity,  is  still  more  powerful  than  that  of  the  sun ;  on  the 
average  two  and  a  half  times  as  great.  Now,  the  moon  crosses  the  celestial 
equator  twice  every  month,  and  at  those  times  her  action  ceases. 


NUTATION.  145 

There  is  still  another  cause  for  variation  in  the  effectiveness  of  the  moon's 

attraction.     As  we  shall  see  hereafter,  she  does  not  move  in  the  ecliptic,  but 

in  a  path  which  cuts  the  ecliptic  at  an  angle  of  about  5°,  at  two  points  called 

the  Nodes;  the  ascending  node  being  the  point  where  she  crosses  the  ecliptic 

_^___ from  south  to  north.     These 

fi  A^jfZ~'  ^^^B  nodes  move  westward  on  the 

^>  j?^<\  ecliptic  (Art.  455),  making 

^s.  the  circuit  once  in  about  nine- 
FlG-  68-  teen  years.  Now,  when  the 

Variation  in  the  Inclination  of  Moon's  Orbit  to  ascending  node  of  the  moon's 

orbit  is  at  B  (Fig.  68),  near 

the  autumnal  equinox  F,  its  inclination  to  the  equator  will  be,  as  the  figure 
shows,  less  than  the  obliquity  of  the  ecliptic  by  about  5°;  i.e.,  it  will  be  only 
about  18°.  On  the  other  hand,  nine  and  a  half  years  later,  when  the  node 
has  backed  around  to  a  point  A,  near  the  vernal  equinox,  the  inclination  of 
the  moon's  orbit  to  the  equator  will  be  nearly  28°.  When  the  node  is  in 
this  position,  the  moon  will  produce  nearly  twice  as  much  precessional 
movement  each  month  as  when  the  node  was  at  B. 

The  precession,  therefore,  is  not  uniform,  but  variable,  almost  ceasing  at 
some  times  and  at  others  becoming  rapid.  This  causes  what  is  called  the 
equation  of  the  equinox. 

214.  Nutation.  —  Not  only  does  the  precessional  force  vary  in 
amount  at  different  times,  but  in  most  positions  of  the  disturbing 
body  with,  respect  to  the  earth's  equator  there  is  a  slight  thwartwise 
component  of  the  force,  tending  directly  to  accelerate  or  retard  the  pre- 
cessional movement  of  the  pole  —  just  as  if  one  should  gently  draw 
the  weight  W  (Fig.  66)  horizontally.     The  consequence  is  what  is 
called  Nutation  or  "nodding."     The  axis  of  the  earth,  instead  of 
moving  smoothly  in  a  circle,  nods  in  and  out  a  little  with  respect  to 
the  pole  of  the  ecliptic,  describing  a  wavy  curve  resembling  that 
shown  in  Fig.  69,  but  with  nearly  1400  indentations  in  the  entire 
circumference  traversed  in  26,000  years. 

215.  We  distinguish  three  of  these  nutations,    (a)  The  Lunar  Nutation, 
depending  upon  the  motion  of  the  moon's  nodes. 

This  has  a  period  of  a  little  less  than  nineteen 
years,  and  amounts  to  9 ".2.  (&)  The  Solar  Nuta- 
tion, due  to  the  changing  declination  of  the  sun. 
Its  period  is  a  year,  and  its  amount  1".2.  (c)  The 
Monthly  Nutation,  precisely  like  the  solar  nutation, 
except  that  it  is  due  to  the  moon's  changes  of  dec- 
lination during  the  month.  It  is,  however,  too 
small  to  be  certainly  measured,  not  exceeding  one- 
tenth  of  a  second.  FlG>  a>  J^utation. 


146  THE   CALENDAR. 

Nutation  was  detected  by  Bradley  in  1728,  but  not  fully  explained  until 
1748. 

Neither  precession  nor  nutation  affects  the  latitudes  of  the  stars,  since  they 
are  not  due  to  any  change  in  the  position  of  the  ecliptic,  but  only'to  displacements 
of  the  earth's  axis.  The  longitudes  alone  are  changed  by  them. 

The  right  ascension  and  declination  of  a  star  are  both  affected. 

216.  The  Three  Kinds  of  Year.  —  In  consequence  of  the  motion 
A  of  the  equinoxes  caused  by  precession,  the  sidereal  year  and  the 
equinoctial  or  "  tropical"  year  do  not  agree  in  length.  Although  the 
sidereal  year  is  the  one  which  represents  the  earth's  true  orbital  revo- 
lution around  the  sun,  it  is  not  used  as  the  year  of  chronology  and 
the  calendar,  because  the  seasons  depend  on  the  sun's  place  in  rela- 
tion to  the  equinoxes.  The  tropical  year  is  the  year  usually  employed, 
unless  it  is  expressly  stated  to  the  contrary.  The  length  of  the 
Sidereal  year  is  365d  6h  9m  9s;  that  of  the  Tropical  year  is  about 
20m  less,  365d  5h  48m  46s. 

The  third  kind  of  year  is  the  anomalistic  year,  which  is  the  time 
from  perihelion  to  perihelion  again.  As  the  line  of  apsides  of  the 
earth's  orbit  moves  always  slowly  towards  the  east,  this  year  is  a  little 
longer  than  the  sidereal.  Its  length  is  365d  6h  13m  48s. 

^  217.  The  Calendar.  —  The  natural  units  of  time  are  the  day,  the 
month,  and  the  year.  The  clay,  however,  is  too  short  for  convenient 
use  in  designating  extended  periods  of  time,  as  for  instance  in 
expressing  the  age  of  a  man.  The  month  meets  with  the  same 
objection,  and  for  all  chronological  purposes,  therefore,  the  year  is 
the  unit  practically  employed.  In  ancient  times,  however,  so  much 
regard  was  paid  to  the  month,  and  so  many  of  the  religious  beliefs 
and  observances  connected  themselves  with  the  times  of  the  new  and 
full  moon,  that  the  early  history  of  the  calendar  is  largely  made  up 
of  attempts  to  fit  the  month  to  the  year  in  some  convenient  way. 
Since  the  two  are  incommensurable,  the  problem  is  a  very  difficult, 
and  indeed  strictly  speaking,  an  impossible,  one. 

In  the  earliest  times  matters  seem  to  have  been  wholly  in  the  hands 
of  the  priesthood,  and  the  calendar  then  was  predominantly  lunar, 
with  months  and  days  intercalated  from  time  to  time  to  keep  the 
seasons  in  place.  The  Mohammedans  still  use  a  purely  lunar  calen- 
dar, having  a  year  of  twelve  lunar  months,  and  containing  alternately 
354  and  355  days.  In  their  reckoning  the  seasons  fall,  of  course, 
continually  in  different  months,  and  their  calendar  gains  about  one 
vear  in  thirty-three  upon  the  reckoning  of  Christian  nations. 


THE   CALENDAR.  147 

218.  The  Metonic  Cycle.  —  Among  the  Greeks  the  discovery  of 
the  so-called  lunar  or  Metonic  cycle  by  Meton,  about  433  B.C.,  con- 
siderably simplified  matters.     This   cycle   consists  of   235  synodic 
months  (from  new  moon  to  new  again),  which  is  very  approximately 
equal  to  19  common  years  of  365J  days. 

235  months  equal  6939d  16h  31m  ;  19  tropical  years  equal  6939d  14h  27m  ; 
so  that  at  the  end  of  the  19  years,  the  new  and  full  moon  recur  again  on  the 
same  days  of  the  year,  and  at  the  same  time  of  day  within  about  two  hours. 
The  calendar  of  the  phases  of  the  moon,  for  instance,  for  1889  is  the  same 
as  for  1870  and  1908,  except  that  intervening  leap-years  may  change  the 
dates  by  one  day. 

The  "  Golden  number  "  of  a  year  is  its  number  in  this  Metonic  cycle,  and 
is  found  by  adding  1  to  the  "  date-number  "  of  the  year  and  dividing  by  19. 
The  remainder,  unless  zero,  is  the  "  golden  number  "  (if  it  comes  out  zero, 
19  is  taken  instead).  Thus  the  golden  number  for  1888  is  found  by  divid- 
ing 1889  by  19,  and  the  remainder  8  is  the  golden  number  of  the  year. 

This  cycle  is  still  employed  in  the  ecclesiastical  calendar  in  finding  the 
time  of  Easter. 

The  still  more  accurate  Callipic  cycle  consists  of  76  years,  or  four  Metonic 
I  cycles,  and  so  takes  account  of  the  leap-years. 

219.  Julian  Calendar.  —  Until  the  time  of  Julius  Caesar  the  Roman 
calendar  seems  to  have  been  based  upon  the  lunar  year  of  twelve 
months,  or  355  days,  and  was  substantially  like  the  modern  Mohamme- 
dan calendar,  with  arbitrary  intercalations  of  months  and  days  made  by 
the  priesthood  and  magistrates  from  time  to  time  in  order  to  bring  it 
into  accordance  with  the  seasons.     In  the  later  days  of  the  Republic, 
the  confusion  had  become  intolerable.    Caesar,  with  the  help  of  the  as- 
tronomer Sosigenes,  whom  he  called  from  Alexandria  for  the  purpose, 
reformed  the  system  in  the  year  45  B.C.,  introducing  the  so-called 
"Julian  calendar,"  which  is  still  used  either  in  its  original  shape  or 
with  a  very  slight  modification.    He  gave  up  entirely  the  attempt  to  co- 
ordinate the  month  with  the  year,  and  adopting  365J  days  as  the  true 
length  of  the  tropical  year,  he  ordained  that  every  fourth  year  should 
contain  an  extra  day,  the  sixth  day  before  the  Kalends  of  March  on  that 
year  being  counted  twice,  whence  the   year  was  called  "  bissextile." 
Before  his  time  the  year  had  begun  in  March  (as  indicated  by  the 
Roman  names  of  the  months,  —  September,  seventh  month  ;  October, 
eighth  month,  etc.),  but  he  ordered  it  to  begin  on  the  1st  of  January, 
which  in  that  year  (45  B.C.)  was  on  the  day  of  the  new  moon  next  fol- 
lowing the  winter  solstice.    In  introducing  the  change  it  was  necessary 
to  make  the  preceding  year  445  days  long,  and  it  is  still  known  in 


148  THE   CALENDAR. 

the  annals  as   "  the  year  of  confusion."     He  also  altered  the  name 
of  the  month  Quintilis,  calling  it  "  July  "  after  himself. 

There  was  some  irregularity  in  the  bissextile  years  for  a  few  years  after 
Caesar's  death,  from  a  misunderstanding  of  his  rule  for  the  intercalary  day  ; 
but  his  successor  Augustus  remedied  that,  and  to  put  himself  on  the  same 
level  with  his  predecessor,  he  took  possession  of  the  month  Sextilis,  calling- 
it  "August";  and  to  make  its  length  as  great  as  that  of  July,  he  robbed 
February  of  a  day. 

From  that  time  on,  the  Julian  calendar  continued  unbrokenly  in  use  until 
1582  ;  and  it  is  still  the  calendar  of  Russia  and  of  the  Greek  Church. 

220.  The  Gregorian  Calendar.  —  The  Julian  calendar  is  not  quite 
correct.     The  true  length  of  the  tropical  year  is  365  days  5  hours 
48  minutes  and  46  seconds,  and  this  leaves  a  difference  of  11  minutes 
and  14  seconds  by  which  the  Julian  calendar  year  is  the  longer,  be- 
ing exactly  365j-  days.     As  a  consequence,  the  date  of  the  equinox 
comes  gradually  earlier  and  earlier  by  about  three  days  in  400  years. 
(400  X  lH$m  =  4493  minutes  =  3d  2h  53m.)     In  the  year  1582,  the 
date  of  the  vernal  equinox  had  fallen  back  10  days  to  the  llth  of 
March,  instead  of  occurring  on  the  21st  of  March,  as  at  the  time  of 
the  Council  at  Nice,  325  A.D.     Pope  Gregory,  therefore,  acting  under 
the  advice  of  the  Jesuit  astronomer,  Clavius,  ordered  that  the  day 
following  October  4  in  the  year  1582  should  be  called  not  the  5th, 
but  the  15th,   and  that  the    rule  for  leap-year  should  be  slightly 
changed  so  as  to  prevent  any  such  future  displacement  of  the  equi- 
nox.    The  rule  now  stands  :   All  years  whose  date-number  is  divisible 
by  four  without  a  remainder  are  leap-years,  unless  they  are  century 
years  (1700,  1800,  etc.).     The  century  years  are  not  leap-years  unless 
their  date-number  is  divisible  by  400,  in  which  case  they  are:    that  is, 
1700,  1800,  and  1900  are  not  leap  years  ;   but  1600,  2000,  and  2400 
are. 

221.  Adoption  of  the  New  Calendar.  —  The  change  was  immedi- 
ately adopted  by  all  Catholic  nations;    but  the  Greek  Church  and 
most  of  the  Protestant  nations,  rejecting  the  Pope's  authority,  de- 
clined to  accept  the  correction.     In  England  it  was  at  last  adopted 
in  the  year  1752,  at  which  time  there  was  a  difference  of  eleven 
days  between  the  two  calendars.     (The  year  1600  was  a  leap-year 
according  to  the  Gregorian  system  as  well  as  the  Julian,  but  1700 
was  not.)     Parliament  in  1751  enacted  that  the  day  following  the 
2d  of  September,  in  the  year  1752,  should  be  called  the  14th  instead 
of  the  3d ;  and  also  that  this  year  (1752),  and  all  subsequent  years, 
should  begin  on  the  first  of  January. 


f '  ' 

BEGINNING    OF    THE    YEAR.  149 

The  change  was  made  under  very  great  opposition,  and  there  were  violent 
riots  in  consequence  in  different  parts  of  the  country,  especially  at  Bristol, 
where  several  persons  were  killed.  The  cry  of  the  populace  was,  "  Give  us 
back  our  fortnight,"  for  they  supposed  they  had  been  robbed  of  eleven  days, 
although  the  act  of  Parliament  was  carefully  framed  to  prevent  any  injustice 
in  the  collection  of  interest,  payment  of  rents,  etc. 

At  present,  since  the  year  1800  was  not  a  leap-year  according  to  the 
Gregorian  calendar,  while  it  was  so  according  to  the  Julian,  the  difference 
between  the  two  calendars  amounts  to  twelve  days  ;  thus  in  Russia  the  19th 
of  August  would  be  reckoned  as  the  7th.  In  Russia,  however,  for  scientific 
and  commercial  purposes  both  dates  are  very  generally  used,  so  that  the  date 
mentioned  would  be  written  August  T7g.  When  Alaska  was  annexed  to  the 
United  States,  its  calendar  had  to  be  altered  by  eleven  days.  (See  Art.  123.) 

Since  February  28,  1900  (Gregorian),  the  difference  has  been  thirteen 
days,  and  will  remain  so  until  A.D.  2100. 

222.  The  Beginning  of  the  Year.— The  beginning  of  the  year  has 
been  at  several  different  dates  in  the  different  countries  of  Europe.  Some 
have  regarded  it  as  beginning  at  Christmas,  the  25th  of  December ;  others, 
on  the  1st  of  January ;  others  still,  on  the  1st  of  March ;  others,  on  the 
25th ;  arid  others  still,  at  Easter,  which  may  fall  on  any  day  between  the 
22d  of  March  and  the  25th  of  April. 

In  England  previous  to  the  year  1752  the  legal  year  commenced  on  the 
25th  of  March,  so  that  when  the  change  was  made,  the  year  1751  necessarily 
lost  its  months  of  January  and  February,  and  the  first  twenty-four  days  of 
March.  Many  were  slow  to  adopt  this  change,  and  it  becomes  necessary, 
therefore,  to  use  considerable  care  with  respect  to  English  dates  which  occur 
in  the  months  of  January,  February,  or  March  about  that  period.  The 
month  of  February,  1755,  for  instance,  would  by  some  writers  be  reckoned 
as  occurring  in  1754.  Confusion  is  best  avoided  by  writing  February  ^ff  i. 

The  so-called  Fictitious  Year,  used  in  the  reduction  of  star-places  (Art. 
797),  begins  at  the  moment  when  the  sun's  mean  longitude  is  280°.  This 
always  occurs  sometime  during  the  31st  of  December. 


First  and  Last  Days  of  the  Year.  —  Since  the  ordinary  civil  year 
consists  of  365  days,  which  is  52  weeks  and  one  day,  the  last  day  of  each 
common  year  falls  on  the  same  day  as  the  first ;  so  that  any  given  date  will 
fall  one  day  later  in  the  week  than  it  did  on  the  preceding  year,  unless  a 
29th  of  February  has  intervened,  in  which  case  it  will  be  two  days  later ; 
that  is,  if  the  3d  of  January,  1889,  falls  on  Thursday,  the  same  date  in  1890 
will  fall  on  Friday. 

223*.  Julian  Period  and  Julian  Epoch.  —  The  Julian  Period  con- 
sists of  7980,  (28  X  19  X  15),  Julian  years,  each  containing  exactly 
365£  clays,  and  its  starting-point  or  "Epoch"  is  January  1,  4713  B.C., 


150 


ABERRATION. 


the  Julian  date  of  January  1,  1  A.D.,  being  J.E.  4714.  It  was 
proposed  by  J.  Scaliger  in  1582  as  a  universal  harmonizer  of  the 
different  systems  of  chronological  reckoning  then  in  use,  and  its 
adoption  has  brought  order  out  of  confusion.  It  is  extensively 
employed  in  astronomical  calculations,  the  date  of  any  phenomenon 
being  expressed  beyond  all  ambiguity  either  by  the  (Julian)  year 
and  day,  or  still  more  simply,  as  day  number  so  and  so  of  the  Julian 
era.  Thus  the  date  of  the  solar  eclipse  of  August  9,  1896,  is  J.E., 
year  6609,  222d  day,  or  simply  Julian-day  2413781,  and  this  is 
perfectly  definite  to  every  astronomer,  whether  he  be  American, 
Russian,  Hebrew,  Mohammedan,  or  Chinese. 

The  number  of  days  between  any  two  events,  even  centuries  apart, 
is  at  once  found  by  merely  taking  the  difference  between  their  Julian- 
day  numbers. 

The  Nautical  Almanacs  give  the  Julian  year  and  the  Julian  day 
corresponding  to  January  1  of  each  year.  Thus 


1896  =  Julian  Year  6609. 

1897  =      "          "     6610. 

1898  =      "          "     6611. 

1899  =      "          "    6612. 


January  1,  1896  =  Julian  Day  2  413560. 
"  1,  1897  =  "  "  2  413926. 
"  1,  1898  =  "  "2  414291. 
"  1,  1899  as  . ««  "2  414656. 


224.  Aberration.  —  Although  in  strictness  the  discussion  of  aberration 
does  not  belong  to  a  chapter  describing  the  earth  and  its  motions,  yet  since  it  is 
a  phenomenon  due  to  the  earth's  motion,  and  affects  the  right  ascension  and 
declination  of  the  stars  in  much  the  same  way  as  do  precession  and  nutation,  it 
may  properly  enough  be  considered  here. 

Aberration  is  the  apparent  displacement  of  a  star,  due  to  the  combi- 
nation of  the  motion  of  light  with  the  motion  of  the  observer. 

The  direction  in  which  we  have  to  point  our  telescope  in  observing 
a  star  is  not  the  same  that  it  would  be  if  the  earth  were  at  rest.  It 
lies  beyond  our  scope  to  show  that  according  to  the  wave  theory  of 
light  the  apparent  direction  of  a 
ray  will  be  affected  by  the  ob- 
server's motion  precisely  in  the 
same  way  (within  very  narrow 
limits)  as  it  would  be  if  light 
consisted  of  corpuscles  shot  off 
from  a  luminous  body,  as  Newton 
supposed.  This  is  the  case,  how- 
ever, as  Doppler  and  others  have 
shown ;  and  assuming  it,  the  ex- 
planation of  aberration  is  easy : 

Suppose  an  observer  Standing  at  FlQ.  70.  _  Aberration  of  a  Baindrop. 


ABEKKATION.  151 

rest  with  a  tube  in  his  hand  in  a  shower  of  rain  where  the  drops  are 
falling  vertically.  If  he  wishes  to  have  the  drops  descend  axially 
through  the  tube  without  touching  the  sides,  he  must  of  course  keep 
it  vertical ;  but  if  he  advances  in  any  direction,  he  must  draw  back  the 
bottom  of  the  tube  by  an  amount  which  equals  the  advance  he  makes 
in  the  time  while  the  drop  is  falling  through  the  tube,  so  that  when 
the  drop  falling  from  B  reaches  A,  the  bottom  of  the  tube  will  be 
there  also ;  i.e.,  he  must  incline  the  tube  forward  by  an  angle  a}  such 
that  tan  a  =  u  -+-  F,  where  F  is  the  velocity  of  the  raindrop  and  u 
that  of  his  own  motion.  In  Fig.  70  BA  =  V  and  AA  =  u. 

225.  Now  take  the  more  general  case : 
Suppose  a  star  sending  us  light  with  a 
velocity  V  in  the  direction  SP,  Fig.  71, 
which  makes  the  angle  0  with  the  line 
of  the  observer's  motion.  He  himself 
is  carried  by  the  earth's  orbital  velocity 
in  the  direction  QP.  In  pointing  the 
telescope  so  that  the  light  may  pass  ex- 
actly along  its  optical  axis,  he  will  have 
•• </Q  u 2ip ! >~  to  draw  back  the  eye-end  by  an  amount 

PIG.  71. -Aberration  of  Light.          $P>  which  3USt  6(luals  the  ^Stance  he 

is  carried  by  the  earth's  motion  during 

the  time  that  the  light  moves  from  0  to  P.  The  star  will  thus 
apparently  be  displaced  towards  the  point  towards  which  he  is  mov- 
ing, the  angle  of  displacement  POQ,  or  a,  being  determined  by 
the  relative  length  and  direction  of  the  two  sides  OP  and  QP  of  the 
triangle  OPQ.  These  sides  are  respectively  proportional  to  the 
velocity  of  light,  F,  and  the  orbital  velocity  of  the  earth,  u. 

The  angle  at  P  being  0,  the  angle  OQP  will  be  (0  —  a),  and  we  shall 
have  from  trigonometry  the  proportion  sin  a  :  sin  (0  —  a)  =  u  :  V. 

To  find  a  from  this,  develop  the  second  term  of  the  proportion  and  divide 
the  first  two  terms  by  sin  a,  which  gives  us 

1  :  sin  0  cot  a  —  cos  Q  =  u  :  V, 
whence 

u  sin  $  cot  a  =  V  +  u  cos  0, 
and 

V  -f-  u  cos  0 

cot  a  =  : . 

u  sin  0 

Taking  the  reciprocal  of  this,  we  have 

tan  a  =  — —  sin  Q. 

V  +  u  cos  0 


152 


ABERRATION. 


The  second  term  in  the  denominator  is  insensible,  since  u  is  only  about  one 
ten-thousandth  of  F,  so  that  we  may  neglect  it.1  This  gives  the  formula 
in  the  shape  in  which  it  ordinarily  appears,  viz., 

tan  a  =  —  sin  #• 


B 

FIG.  72. 
Aberrational  Orbit  of  a  Star. 


The  value  of  a  (denoted  by  a0)  which  obtains  when  0  =  90°  and 
sin  6  =  unity,  is  called  the  Constant  of  Aberration. 

The  value  of  this  constant,  adopted  by  the  Astronomical  Confer- 
ence at  Paris  in  1896,  is  2O.47,f  but  is  still  uncertain  by  at  least 
0".02  or  0".03.  Aberration  was  discov- 
ered and  explained  by  Bradley,  in  1726. 

226.  The  Effect  of  Aberration  upon  the 
Apparent  Places  of  the  Stars.  —  As  the 
earth  moves  in  an  orbit  nearly  circular, 
and  with  a  velocity  so  nearly  uniform  that 
we  may  for  our  present  purpose  disregard 
its  variations,  it  is  clear  that  a  star  at  the 
pole  of  the  ecliptic  will  be  always  displaced 
by  the  same  amount  of  20 ".5,  but  in  a  di- 
rection continually  changing.  It  must, 
therefore,  appear  to  describe  a  little  circle 
41"  in  diameter  during  the  year,  as  shown  in  Fig.  72.  Now  the 
direction  of  the  earth's  orbital  motion  is  always  in  the  plane  of  the 
ecliptic,  and  towards  the  right  hand  as  we  stand  facing  the  sun. 
At  the  vernal  equinox,  therefore,  we  are  moving  toward  the  point 
of  the  ecliptic,  which  is  90°  west  of  the  sun,  i.e.,  towards  the  winter 
solstitial  point,  and  the  star  is  then  displaced  in  that  direction. 
Three  months  later  the  star  will  be  displaced  in  a  line  directed  to- 
wards the  vernal  equinox,  and  so  on.  The  earth,  therefore,  so  to 
speak,  drives  the  star  before  it  in  the  aberrational  orbit,  keeping  it 
just  a  quarter  of  a  revolution  ahead  of  itself. 

A  star  on  the  ecliptic  simply  appears  to  oscillate  back  and  forth 
in  a  straight  line  41"  long. 

Generally,  in  any  latitude  whatever,  the  aberrational  orbit  is  an 
ellipse,  having  its  major  axis  parallel  to  the  ecliptic,  and  always  41" 

1  The  velocity  of  light,  according  to  the  latest  determinations  of  Newcomb  and 
Michelson,  is  299860  kilometers  ±  30  kilometers  (which  equals  186330  miles  ±  20 
miles).  The  mean  velocity  of  the  earth  in  its  orbit,  if  we  assume  the  solar  paral- 
lax to  be  8". 8,  is  29.77  kilometers,  or  18.50  miles;  this  makes  the  constant  of 
aberration  20".478,  a  little  larger  than  that  given  in  the  text. 


ABERRATION.  153 

long,  while  its  minor  axis  is  41"  X  sin  (3,  ft  being  the  star's  latitude, 
or  distance  from  the  ecliptic. 

It  is  worth  noting  that  since  the  "  hodograph  " 1  of  the  earth's  orbital 
motion,  as  shown  by  Hamilton,  is  an  exact  circle,  the  aberrational  orbit  is 
also  a  circle,  notwithstanding  the  eccentricity  of  the  earth's  orbit,  and 
the  variations  in  its  velocity;  but  the  mean  place  of  the  star  (i.e.,  the  place 
it  would  occupy  if  there  were  no  aberration)  is  not  exactly  in  the  centre  of 
this  circle. 

226*.  Diurnal  Aberration.  —  The  motion  of  an  observer  due  to  the 
earth's  rotation  also  produces  a  slight  effect  known  as  the  diurnal  aberration. 
Its  "constant"  is  only  0".31  for  an  observer  situated  at  the  equator;  any- 
where else  it  is  0".31  cos  <j>,  <£  being  the  latitude  of  the  observer. 

For  any  given  starlit  is  a  maximum  when  the  star  is  crossing  the  merid- 
ian, and  then  its  whole  effect  is  slightly  to  increase  the  right  ascension  by  an 
amount  given  by  the  formula,  Aa  =  0".31  cos  <£  sec  &,  8  being  the  star's 
declination. 


EXERCISES  ON  CHAPTER  VI. 

-  1.    What  is  the  meridian  altitude  of  the  sun  at  Princeton  (Lat.  40°  21') 
on  the  day  of  the  summer  solstice  ? 

-  2.    What  is  the  sun's  approximate  right  ascension  at  that  time  ? 

^  3.   On  what  days  during  the  year  wiU  the  sun's  right  ascension  be 
approximately  an  even  hour  (i.e.,  0  h.  2  h.  4  h.,  etc.)  ? 

4.  On  what  days  will  it  be  an  odd  hour? 

5.  What  is  the  approximate  sidereal  time  at  10  P.M.  on  May  12? 

Ans.    13  h.  26  min. 

-  -  0.    At  what  time  will  Arcturus  (R.  A.  =  14  h.  10  min.)  come  to  the 
meridian  on  August  1  ? 

Ans.    About  5  h.  26  min.  P.M. 

7.  About  what  time  of  night  is  Mizar  (R.  A.  =  13  h.  20  min.)  vertically 
under  the  pole  on  October  10  ? 

Ans.    Midnight. 

^  8.   In  what  latitude  has  the  sun  a  meridian  altitude  of  80°  on  June  21  ? 

Ans.    +33°27/. 

1  The  hodograph  of  an  orbit  is  a  curve  in  which  the  direction  of  the  radius 
vector  at  each  point  is  the  direction  of  the  orbital  motion  at  the  corresponding 
point  of  the  orbit,  and  its  length  is  proportional  to  the  velocity. 


154  EXERCISES. 

9.  What  are  the  longitude  and  latitude  (celestial)  of  the  north  celestial 
pole? 

10.  What  are  the  right  ascension  and  declination  of  the  north  pole  of 
the  ecliptic? 

11.  What  are  the  greatest  and  least  angles  made  by  the  ecliptic  with  the 
horizon  at  New  York  (Lat.  40°  43')  ? 

Ans.    (90°  -40°  43')  ±  23°  27'. 

12.  Does  the  vernal  equinox  always  occur  on  the  same  day  of  the  month  ? 
If  not,  why  not  ?     And  how  much  can  the  date  vary  ? 

13.  Will  the  ephemeris  of  the  sun  for  one  year  be  correct  for  every  other 
year,  and,  if  not,  how  much  can  it  be  in  error  ? 

Ans.  A  difference  of  one  and  three-quarters  days  motion  of  the  sun  is 
possible;  as,  for  instance,  between  1897  and  1903,  the  leap-year  being 
omitted  in  1900. 

14.  When  the  sun  is  in  the  sign  of  Cancer  in  what  constellation  is  he  ? 

15.  What  obliquity  of  the  ecliptic  would  reduce  the  width  of  the  tem- 
perate zone  to  zero  ? 

16.  How  long  before  the  earth  will  be  in  perihelion  on  July  1,  instead 
of  January  1  as  at  present?     (See  Arts.  199,  206  and  216.) 

17.  What  will  be  the  Russian  date  corresponding  to  February  28,  1900, 
in  our  calendar  ?     What  corresponding  to  May  1  of  the  same  year  ? 

18.  When  the  equation  of  time  is  16  min.,  as  it  is  on  November  1,  how 
does  the  forenoon  from  sunrise  till  12  o'clock  compare  in  length  with  the 
afternoon  from  12  o'clock  till  sunset? 

19.  Why  do  the  afternoons  begin  to  lengthen  about  December  8,  a 
fortnight  before  the  winter  solstice  ? 

20.  There  were  five  Sundays  in  February,  1880.     The  same  thing  has 
not  occurred  since,  and  will  not  until  —  when  ? 

Ans.   1920. 

NOTE  TO  ART.  182. 

The  difference  between  the  radius-vector  of  the  ellipse,  and  that  of  the 
eccentric  circle  proposed  by  Hipparchus  for  the  orbit  of  the  earth,  is  so  small 
that  the  method  given  in  the  text  would  not  practically  suffice  to  discriminate 
between  them.  But  the  investigation  of  Newton  (Arts.  421  and  1006)  shows 
that  the  orbit  must  be  elliptical  like  that  of  the  other  planets. 


THE   MOON.  155 


CHAPTER  VII. 

THE   MOON  :      HER    OEBITAL    MOTION    AND    VARIOUS    KINDS   OF 
MONTH.  —  DISTANCE  AND  DIMENSIONS,  MASS,  DENSITY,  AND 

GRAVITY.  —  ROTATION  AND  ITERATIONS.  —  PHASES. LIGHT 

AND  HEAT.  —  PHYSICAL  CONDITION  AND  INFLUENCES  EX- 
ERTED ON  THE  EARTH.  —  TELESCOPIC  ASPECT.  —  SURFACE 
AND  POSSIBLE  CHANGES  UPON  IT. 

227,  WE  pass  next  to  a  consideration  of  our  nearest  neighbor  in 
the  celestial  spaces,  the  moon,  which  is  a  satellite  of  the  earth  and 
accompanies  us  in  our  annual  motion  around  the  sun.     She  is  much 
smaller  than  the  earth,  and,  compared  with  most  of  the  other  heav- 
enly bodies,  a  very  insignificant  affair  ;    but  her  proximity  makes 
her  far  more  important  to  us  than  any  of  them  except  the  sun.  The 
very  beginnings  of  Astronomy  seem  to  have  originated  in  the  study 
of  her  motions  and  in  the  different  phenomena  which  she  causes, 
such  as  the  eclipses  and  tides ;  and  in  the  development  of  modern 
theoretical  astronomy  the  lunar  theory  with  the  problems  it  raises 
has  been  perhaps  the  most  fertile  field  of  invention  and  discovery. 

228.  Apparent  Motion  of  the  Moon. — Even  superficial  observation 
shows  that  the  moon  moves  eastward1  among  the  stars  every  night, 
completing  her  revolution  from  star  to  star  again  in  about  27£  days. 
In  other  words,  she  revolves  around  the  earth  in  that  time;    or, 
more  strictly  speaking,  they  both  revolve  about  their  common  centre 
of  gravity.     But  the  moon  is  so  much  smaller  than  the  earth  that 
this  centre  of  gravity  is  situated  within  the  ball  of  the  earth  on  the 
line  joining  the  centres  of  the  two  bodies  at  a  point  about  1100  miles 
below  its  surface. 

As  the  moon  moves  eastward  so  much  faster  than  the  sun,  which 
takes  a  year  to  complete  its  circuit,  she  every  now  and  then,  at 
the  time  of  the  new  moon,  overtakes  and  passes  the  sun ;  and  as  the 
phases  of  the  moon  depend  upon  her  position  with  reference  to  the 
sun,  this  interval  from  new  moon  to  new  moon  is  what  we  ordinarily 
understand  as  the  month. 

1  She  moves  in  declination,  also,  —  alternately  north  and  south  (Art.  234). 


156  THE   MOON. 

229.  Sidereal  and  Synodic  Revolutions,  —  The  SIDEREAL  revolu- 
tion of  the  moon  is  the  time  occupied  in  passing  from  a  star  to  the  same 
star  again,  as  the  name  implies.    It  averages  27d  7h  43m  118.55  ±08.03, 
or  27d.32166,  but  varies   as   much  as  3  hours   on.  account  of  the 
eccentricity  of  its  orbit  and  "perturbations." 

The  moon's  daily  motion  among  the  stars  equals  360° -f- 27. 321 66, 
or  13°  11'  (nearly). 

The  SYNODIC  revolution  is  the  interval  from  new  moon  to  new  moon 
again,  or  from  full  to  full.  It  varies  about  13h  on  account  of  the 
eccentricity  of  the  moon's  orbit  and  of  that  of  the  earth  around  the 
sun,  but  its  mean  value  is  29d  12h  44m  28.86  ±  08.03,  or  29d.53059 ; 
and  this  is  the  ordinary  month.  (The  word  synodic  is  derived  from 
the  Greek  a-vv  and  6Sos,  and  has  nothing  to  do  with  the  nodes  of  the 
moon's  orbit.  The  word  is  syn-odic,  not  sy-nodic.} 

A  synodical  revolution  is  longer  than  the  sidereal,  because  during 
each  sidereal  month  of  27.3  days  the  sun  has  advanced  among  the 
stars,  and  must  be  overtaken. 

230.  Elongation,  Syzygy,  etc.  —  The  angular  distance  of  the  moon 
from  the  sun  is  called  its  Elongation.     At  new  moon  it  is  zero,  and 
the  moon  is  then  said  to  be  in  "  Conjunction.'"     At  full  moon  it  is 
180°,  and  the  moon  is  then  in  u  Opposition."     In   either  case   the 
moon  is  said  to  be  in  "  Syzygy"  (a-vv  £vy6v).     When  the  elongation 
is  90°,  as  at  the  half-moon,  the  moon  is  in  "  Quadrature." 

231.  Determination  of    the  Moon's  Sidereal  Period.  —  This  is 
effected  directly  by  observations  of  the  moon's  right  ascension  and 
declination  (with  the  meridian  circle) ,  kept  up  systematically  for  a 
sufficient  time. 

If  it  were  not  for  the  so-called  "secular  acceleration"  of  the 
moon's  motion  (Arts.  459-461),  an  exceedingly  accurate  determina- 
tion of  the  moon's  synodic  period  could  be  obtained  by  comparing 
ancient  eclipses  with  modern. 

The  earliest  authentically  recorded  eclipse  is  one  that  was  observed 
at  Nineveh  in  the  year  763  B.C.  between  9  and  10  o'clock  on  the 
morning  of  June  15th. 

By  comparing  this  eclipse  with  (say)  the  eclipse  of  August,  1887, 
we  have  an  interval  of  more  than  30000  months,  and  so  an  error  of 
ten  hours  even,  in  the  observed  time  of  the  Nineveh  eclipse,  would 
make  only  about  one  second  in  the  length  of  the  month.  But  the 
month  is  a  little  shorter  now  than  it  was  2000  years  ago. 


SIDEREAL   AND    SYNODIC   PEEIODS.  157 

232.    Relation  of  Sidereal  and  Synodic  Periods.  —  The  fraction  of 

a  revolution  described  by  the  moon  in  one  day  equals  —  ,  M  being  the 

1 

length    of    the    sidereal    month.      In    the    same    way  —  represents 

lii 

the  earth's  daily  motion  in  its  orbit,  E  being  the  length  of  the  year. 
The  difference  of  these  two  equals  the  fraction  of  a  revolution  which 
the  moon  gains  on  the  sun  during  one  day.  In  a  synodic  month,  $,  it 

gains  one  whole  revolution,  and  therefore  must  gain  each  day  -  of  a 
revolution  ;  so  that  we  have  the  equation 


__=  . 

M     E     S' 
or,  substituting  the  numerical  values  of  E  and  S, 


M     365.25635      29.53059* 
whence  we  derive  the  value  of  M. 

Another  way  of  looking  at  it  is  this  :  In  a  year  there  must  be  exactly  one 
more  sidereal  revolution  than  there  are  synodic  revolutions,  because  the  sun 
completes  one  entire  circuit  in  that  time.  Now  the  number  of  synodic  revo- 
lutions in  a  year  is  given  by  the  fraction 

5§§i=  12.369+. 
S 

There  will  therefore  be  13.369  sidereal  revolutions  in  the  year,  and  the 
length  of  one  sidereal  revolution  equals  365^  days  divided  by  this  number 
13.369,  which  will  be  found  to  give  the  length  of  the  sidereal  revolution  as 
before. 

233.  Moon's  Path  among  the  Stars.  —  By  observing  with  the  me- 
ridian circle  the  right  ascension  and  declination  of  the  moon  daily 
during  the  month,  just  as  in  the  case  of  the  sun,  we  obtain  the  posi- 
tion of  the  moon  for  each  day,  and  joining  the  points  thus  found,  we 
can  draw  the  path  of  the  moon  in  the  sky.  It  is  a  great  circle, 
cutting  the  ecliptic  in  two  points  called  the  nodes,  at  an  angle  which 
ranges  from  4°  57'  to  5°  20',  the  mean  being  about  5°  08'. 

We  say  the  path  is  found  to  be  a  great  circle.  This  must  be  taken 
with  some  reservation,  since  at  the  end  of  the  month  the  moon  never 
returns  precisely  to  the  position  it  occupied  at  the  beginning,  owing 


158  THE   MOON. 

to  the  regression  of  the  nodes  and  other  so-called  "  perturbations," 
which  will  be  discussed  hereafter. 

234.  Moon's  Meridian  Altitude.  —  Since  the  moon's  orbit  is  inclined 
to  the  ecliptic  5°  8',  its  inclination  to  the  equator  varies  from  28°  36'  (23°  28' 
+  5°  8'),  when  the  moon's  ascending  node  is  the  vernal  equinox,  to  18°  20', 
when,  9|  years  later,  the  same  node  is  at  the  autumnal  equinox.     In  the  first 
case  the  moon's   declination  will   change   during  the  month  by  57°  12', 
from  -  28°  36'  to  +  28°  36'.     In  the  other  case  it  will  change  only  by  36°  40'. 
Since  the  range  of  the  moon's  meridian  altitude  is  the  same  as  that  of  its 
declination,  the  difference  becomes  striking. 

235.  Interval  between  Moon's  Transits.  —  On  the  average  the  moon 
gains  12°  llr.4  on  the  sun  daily,  so  that  it  comes  to  the  meridian  about 
51  minutes  of  solar  time  later  each  day. 

To  find  the  mean  interval  between  the  successive  transits  of  the 
moon  we  may  use  the  proportion 

(360° -12°  11'.4)  :  360°  =  24h  :  a; ;  whence  x  —  24h  50m.6. 

The  variations  of  .the  moon's  motion  in  right  ascension,  which  are 
very  considerable  (much  greater  than  in  the  case  of  the  sun) ,  cause 
this  interval  to  vary  from  24h  38m  to  25h  06m. 

236.  The  Daily  Retardation  of  the  Moon's  Rising  and  Setting.  — 

The  average  daily  retardation  of  the  moon's  rising  arid  setting  is, 
of  course,  the  same  as  that  of  her  passage  across  the  meridian, 
viz.,  51m;  but  the  actual  retardation  of  rising  is  subject  to  very 
much  greater  variations  than  those  of  the  meridian  passage,  being 
affected  by  the  moon's  changes  in  declination  as  well  as  by 
the  inequalities  of  her  motion  in  right  ascension.  When  the  moon 
is  very  far  north,  having  her  maximum  declination  of  28°  36',  she 
will  rise  in  our  latitudes  much  earlier  than  when  she  is  farther 
south,  though  having  the  same  right  ascension. 

In  the  latitude  of  New  York  the  least  possible  daily  retardation 
of  moon-rise  is  23  minutes,  and  the  greatest  is  1  hour  and  17 
minutes.  In  higher  latitudes  the  variation  is  greater  yet. 

237.  Harvest  and  Hunter's  Moons.  —  The  variations  in  the  retarda- 
tion of  the  moon's  rising  attract  most  attention  when  they  occur  at  the  time 
of  the  full  moon.     When  the  retardation  is  at  its  minimum,  the  moon  rises 
soon  after  sunset  at  nearly  the  same  time  for  several  successive  evenings ; 
whereas,  when  the  retardation  is  greatest,  the  moon  appears  to  plunge  nearly 


HARVEST    AND   HUNTER'S    MOONS.  159 

vertically  below  the  horizon  by  her  daily  motion.  When  the  full  moon 
occurs  at  the  time  of  the  autumnal  equinox,  the  moon  itself  will  be  near 
the  first  of  Aries. 

Now,  as  will  be  seen  by  reference  to  Fig.  73,  the  portion  of  the  ecliptic 
near  the  first  of  Aries  makes  a  much  smaller  angle  with  the  eastern  horizon 
than  the  equator. 

[The  line  UN  is  the  horizon,  E  being  the  east  point  —  the  figure  being 
drawn  to  represent  a  celestial  globe,  as  if  the  observer  were  looking*  at  the 
eastern  side  of  the  celestial  sphere  from  the  outside.'] 

EQis  the  equator.  Now,  when  the  autumnal  equinoctial  point  or  first  of 
Libra  is  on  the  horizon  at  E,  the  position  of  the  ecliptic  will  be  that  repre- 
sented by  ED ;  more  steeply  inclined  to  the  horizon  than  EQ  is,  by  the 
angle  QED,  23i°.  But  when  the  first  of  Aries  is  at  E,  the  ecliptic  will  be  in 


FIG.  73.  —  Explanation  of  the  Harvest  Moon. 

the  position  JJ'.  And  if  the  ascending  node  of  the  moon's  orbit  happens 
then  to  be  near  the  first  of  Aries,  the  moon's  path  will  be  MM'. 

Accordingly,  when  the  moon  is  in  Aries,  it,  so  to  speak,  coasts  along  the 
eastern  horizon  from  night  to  night,  its  time  of  rising  not  varying  very 
much  ;  and  this,  when  it  occurs  near  the  full  of  the  moon,  gives  rise  to  the 
phenomenon  known  as  the  harvest  moon,  the  harvest  moon  being  the  full  moon 
nearest  to  the  autumnal  equinox.  The  full  moon  next  following  is  called  the 
hunter's  moon. 

In  .Norway  and  Sweden,  under  these  circumstances,  the  moon's  orbit  may 
actually  coincide  with  the  horizon,  so  that  she  will  rise  at  absolutely  the  same 
time  for  a  considerable  number  of  successive  evenings. 

238.  The  Moon's  Orbit.  —  As  in  the  case  of  the  sun,  the  observa- 
tion of  the  moon's  path  in  the  sky  gives  no  information  as  to  the  real 
size  of  its  orbit ;  but  its  form  may  be  found  by  measuring  the  appar- 
ent diameter  of  the  moon,  which  ranges  from  33'  30"  to  29'  21"  at 
different  points.  The  orbit  turns  out  to  be  an  ellipse  like  the  orbit 
of  the  earth,  but  with  an  eccentricity  more  than  three  times  as  great 


160 


THE   MOON. 


-  about  TJ¥  on  the  average,  but  varying  from  T*¥  to  fa  on  account 
of  perturbations. 

The  extremities  of  the  major  axis  of  the  moon's  orbit  are  called 
the  perigee  and  apogee  (from  irf.pl  yrj  and  0.71-0  yfj). 

The  line  of  apsides,  which  passes  through  these  two  points,  moves 
around  towards  the  east  once  in  about  nine  years,  also  on  account  of 
perturbations. 

239,  Distance  and  Parallax  of  the  Moon.  —  These  can  be  found  in 
several  ways,  of  which  the  simplest  is  the  following  :  At  two  ob- 
servatories B  and  C  (Fig.  74) 
on,  or  very  nearly  on,  the  same 
meridian  and  very  far  apart  (in 
the  northern  and  southern  hemi- 
spheres if  possible ;  Greenwich 
and  the  Cape  of  Good  Hope,  for 
instance)  let  the  moon's  zenith 
distance  ZBM  and  Z'CM  be  ob- 
served simultaneously  with  the 
meridian  circle.  This  gives1  in 

FIG.  74.  -Determination  of  the  Moon's  Distance.    the    quadrilateral    BOCM    the 

two  angles  at  B  and  C,  each  of 

which  is  the  supplement  of  the  geocentric  zenith  distance.  The 
angle  at  the  centre  of  the  earth,  BOC,  is  the  difference  of  the  geo- 
centric latitudes  and  is  known  from  the  geographical  positions  of 
the  two  observatories.  Knowing  the  three  angles  in  the  quadrilat- 
eral, the  fourth  at  M  is  of  course  known.  The  sides  BO  and  CO 
are  known,  being  radii  of  the  earth  ;  so  that  we  can  solve  the  whole 
quadrilateral  by  a  simple  trigonometrical  process. 

First  find  from  the  triangle  BOC  the  partial  angles  OCB  and  OBC,  and 
the  side  BC.  Then  in  the  triangle  BCM  we  have  BC  and  the  two  angles 
CBM  and  MCB,  from  which  we  can  find  the  two  sides  BM  and  CM. 
Finally,  in  the  triangle  OBM,  we  now  know  the  sides  OB  and  BM  and  the 
included  angle  OBM,  so  that  the  side  OM  can  be  computed,  which  is  the 
distance  of  the  moon  from  the  earth's  centre.  Knowing  this,  the  horizontal 
parallax  KMO,  or  the  semi-diameter  of  the  earth  as  seen  from  the  moon, 
follows  at  once  from  the  right-anlged  triangle  OKM. 

The  moon's  parallax  can  also  be  deduced  from  observations  at  a  single 
station  on  the  earth,  but  not  so  simply.  If  she  did  not  move  among  the 
stars,  it  would  be  very  easy,  as  all  we  should  have  to  do  would  be  to  compare 
her  apparent  right  ascension  and  declination  at  different  points  in  her  diur- 


1  By  correcting  the  observed  zenith-distance  for  the  angle  of  the  vertical 
(Art.  156). 


THE   MOON'S    PARALLAX   AND   DISTANCE. 


161 


nal  circle.  Near  the  eastern  horizon  the  parallax  (always  depressing  an  ob- 
ject) increases  her  right  ascension  ;  at  setting,  vice  versa.  On  the  meridian 
the  declination  only  is  affected.  But  the  motion  of  the  moon  must  be  al- 
lowed for,  as  the  observations  to  be  compared  are  necessarily  separated  by 
considerable  intervals  of  time,  and  this  complicates  the  calculation. 

A  third,  and  a  very  accurate,  method  is  by  means  of  occultations  of 
stars,  observed  at  widely  separated  points  on  the  earth.  These  occultations 
furnish  the  moon's  place  with  great  accuracy,  and  so  determine  the  paral- 
lax very  precisely;  but  the  calculation  is  not  very  simple,  as  the  moon's 
motion  in  this  case  also  enters  into  it,  since  the  observations  cannot  be 
simultaneous. 

240.  The  Distance  of  the  Moon  is  continually  changing  on  account 
of  the  eccentricity  of  its  orbit,  varying  all  the  way,  according  to  Nei- 
son,  between  252,972  and  221,614  miles;  the  mean  distance  being 
238,840  miles,  or  60.27  times  the  equatorial  radius  of  the  earth. 
The  mean  parallax  of  the  moon  is  57'  2",  subject  to  a  similar  per- 
centage of  change.  This  value  of  the  parallax,  it  will  be  noted, 
indicates  that  the  earth,  as  seen  from  the  moon,  has  a  diameter  of 
nearly  2°. 

Knowing  the  size  of  the  moon's  orbit  and  the  length  of  the  month, 
the  velocity  of  her  motion  around  the  earth  is  easily  calculated.  It 
comes  out  2288  miles  per  hour,  or  about  3350  feet  a  second. 


FIG.  75.  —  Moon's  Path  with  Reference  to  the  Sun. 


Fie.  76. 


FIG.  77. 


False  Representations  of  Moon's  Motions. 


941.  Form  of  the  Moon's  Orbit  with  Reference  to  the  Sun. — 
While  the  moon  moves  in  a  small  elliptical  orbit  around  the  earth,  it 
also  moves  around  the  sun  in  company  with  the  earth.  This  common 


162  THE   MOON. 

motion  of  the  moon  and  earth,  of  course,  does  not  affect  their  relative 
motion  ;  but  to  an  observer  outside  the  system  the  moon's  motion 
around  the  earth  would  only  be  a  verj7  small  component  of  the  moon's 
movement  as  seen  by  him. 

The  distance  of  the  moon  from  the  earth,  239,000  miles,  is  very 
small  compared  with  that  of  the  earth  from  the  sun,  93,000,000  miles 
—  being  only  about  ^^  part.  The  speed  of  the  earth  in  its  orbit 
around  the  sun  is  also  more  than  thirty  times  faster  than  that  of  the 
moon  in  its  orbit  around  the  earth,  so  that  for  the  moon  the  result- 
ing path  in  space  is  one  which  is  always  concave  towards  the  sun, 
as  shown  in  Fig.  75.  It  is  not  like  Figs.  76  and  77,  as  often  rep- 
resented. If  we  represent  the  orbit  of  the  earth  by  a  circle  with  a 
radius  of  100  inches  (8  feet  4  inches) ,  the  moon  would  only  move 
out  and  in  a  quarter  of  an  inch,  crossing  the  circumference  twenty- 
five  times  in  going  once  around  it. 

242.  Diameter  of  the  Moon.  —  The  mean  apparent  diameter  of  the 
moon  is  31'  7".     This  gives  it  a  real  diameter  of  2163  miles  (plus  or 
minus  one  mile),  which  equals  0.273  of  the  earth's  diameter.     Since 
the  surfaces  of  globes  are  as  the  squares  of  their  diameters,  and  their 
volumes  as  their  cubes,  this  makes  the  surface  of  the  moon  0.0747  of 
the  earth's  (between  y1^  and  y1^)  ;  and  the  volume  0.0204  of  the  earth's 
volume  (almost  exactly  ^)  ;  that  is,  it  would  take  49  balls  each  as 
large  as  the  moon  in  bulk  to  make  a  ball  of  the  size  of  the  earth. 

243.  Mass  of  the  Moon. — This  is  about  -^  of  the  earth's  mass, 
different  authorities  giving  the  value  from  -^  to  -fa.     It  is  not  easy 
to  determine  it  with  accuracy.     In  fact,   though   the   moon   is   the 
nearest  of  all  the  heavenly  bodies,  it  is  more  difficult  to  "  weigh"  her 
than  to  weigh  Neptune,  although  he  is  the  most  remote  of  the  planets. 

There  are  four  ways  of  approaching  the  problem :  (1)  (perhaps 
easiest  to  understand)  by  finding  the  position  of  the  common  centre  of 
gravity  of  the  earth  and  moon  with  reference  to  the  centre  of  the  earth. 
Since  it  is  this  common  centre  of  gravity  of  the  two  bodies  which 
describes  around  the  sun  the  ellipse  which  we  have  called  the 
earth's  orbit,  and  since  the  earth  and  moon  revolve  around  this 
common  centre  of  gravity  once  a  month,  it  follows  that  this  monthly 
motion  of  the  earth  causes  an  alternate  eastward  and  westward 
displacement  of  the  sun  in  the  sky,  which  can  be  measured.  At 
the  time  of  the  new  and  full  moon  this  displacement  is  zero,  the 
centre  of  gravity  being  on  the  line  which  joins  the  earth  and  sun ; 


THE    MOON  S    DENSITY. 


163 


(B) 


but  when  the  moon  is  at  quadrature  (that  is,  90°  from  the  sun,  as  at 

the  time  of  half -moon),  the  sun 

is   apparently  displaced   in  the  E^  E^ 

sky  towards  the  moon,  as  is  evi-  ^    l-^J  /~rN    v~~  '     MI 

dent  from  Fig.  78.     It  will  be 

about  6".4  east  of  its  mean  place 

at  the  first  quarter  of  the  moon, 

Fig.  78  (B),  and  as  much  west 

at  the  time  of  the  last  quarter, 

Fig.  78  (^4) ;  (i.e.,  when  the  angle 

MGS  is  90°,  the  angle  MCS  is 

always    less   than   90°   by   6".4, 

which  is  therefore  the  value  of 

the  angle  CSG).     Now  since  the 

parallax  of  the  sun  (which  is  the 

earth's  semi-diameter  seen  from 

the    sun  —  the   angle    CSK)    is 

about  8  ".8,  it  follows   that  the 

distance  of  the  centre  of  gravity 

of  the  earth  and  moon  from  the 

centre  of  the  earth  is  the  fraction 

||  of  the  earth  radius,  or  about 

2880  miles.     This  is  just  about 

~-5  of  the  distance  from  the  earth  to  the  moon ;  whence  we  conclude 

that  the  mass  of  the  earth  is  81.5  times  that  of  the  moon. 


FIG.  78. 

Apparent  Displacement  of  Sun  at  First  and 
Third  Quarters  of  the  Month. 


244.  (2)  A  second  method  is  by  comparing  the  moon's  actual  period 
with  the  computed  period  which  a  single  particle  at  the  moon's  distance  from  the 
earth  ought  to  have,  according  to  the  known  force  of  gravity  of  the  earth,  as 
determined  by  pendulum  experiments.  The  explanation  of  this  method 
cannot  be  given  until  we  have  further  studied  the  motion  of  bodies  under 
the  law  of  gravitation.  It  is  not  susceptible  of  great  accuracy. 

(3)  Still  another  method  is  by  comparing  the  tides  produced  by  the  moon 
with  those  produced  by  the  sun.     This  gives  us  the  mass  of  the  moon  as  com- 
pared with  that  of  the  sun ;   and  the  mass  of  the  sun  compared  with  that  of 
the  earth  being  known,  it  gives  us  ultimately  the  mass  of  the  moon  compared 
with  that  of  the  earth. 

(4)  The  ratio  of  the  moon's  mass  to  the  sun's  can  also  be  computed  from 
the  nutation  of  the  earth's  axis.     (See  Chap.  XIII.) 


245.     No  other  satellite  is  nearly  as  large  as  the  moon,  in  comparison 
with  its  primary  planet.     The  earth  and  moon  together,  as  seen  from  a  dis- 


164  THE   MOON. 

tant  star,  are  really  in  many  respects  more  like  a  double  planet  than  like  a 
planet  and  satellite,  as  ordinarily  proportioned  to  each  other.  At  a  time, 
for  instance,  when  Venus  happens  to  be  near  the  earth,  at  a  distance  of 
about  twenty-five  millions  of  miles,  the  earth  to  her  would  appear  about 
twice  as  bright  as  Venus  at  her  best  does  to  us  ;  and  the  moon  would  be 
about  as  bright  as  Sirius,  at  a  distance  of  about  half  a  degree  from  the  earth. 

246.  Density  and  Superficial  Gravity  of  the  Moon.  —  Since  the 
density  of  a  body  is  equal  to  —  -  -  ,  the  density  of  the  moon  as 
compared  with  that  of  the  earth  is  found  from  the  fraction 

i          0.0124 
L    f\-p  _ 

?V       0.0204' 

This  makes  the  moon's  density  0.61  of  the  earth's  density,  or  about 
3T%  the  density  of  water  —  somewhat  above  the  average  density  of 
the  rocks  which  compose  the  crust  of  the  earth. 

This  small  density  of  the  moon  is  not  surprising,  nor  at  all  inconsistent 
with  the  belief  that  it  once  formed  part  of  the  same  mass  with  the  earth, 
since  if  such  were  the  case,  the  moon  was  probably  formed  by  the  separation 
of  the  outer  portions  of  that  mass,  which  would  be  likely  to  have  a  smaller 
specific  gravity  than  the  rest. 

247.  The  superficial  gravity,  or  the  attraction  of  the  moon  for 
bodies  at  its  own  surface,  may  be  found  by  the  equation 


in  which  gf  signifies  the  superficial  gravity  of  the  moon,  g  is  the 
force  of  gravity  of  the  earth,  while  m  and  r  are  the  mass  and  radius 
of  the  moon  as  compared  with  those  of  the  earth.  This  gives  us 

0.0124 


or  (very  approximately)  g'  equals  one-sixth  of  g  ;  that  is,  a  body  which 
weighs  six  pounds  on  the  earth's  surface  would  at  the  surface  of  the 
moon  weigh  only  one  (in  a  spring  balance).  A  man  on  the  moon 
could  jump  six  times  as  high  as  he  could  on  the  earth  and  could  throw 


..'• 

ROTATION    OF    THE    MOON.  165 

a  stone  six  times  as  far.  This  is  a  fact  to  be  remembered  in  connec- 
tion with  the  enormous  scale  of  the  surface-structure  of  the  moon. 
Volcanic  forces,  for  instance,  upon  the  moon  would  throw  the  ejected 
materials  to  a  vastly  greater  distance  there  than  on  the  earth. 


n 
L 


248,  Eotation  of  the  Moon.  —  The  moon  rotates  on  its  axis  once  a 
month,  in  precisely  the  same  time  as  that  occu- 
pied  by  its  revolution  around  the  earth.  In  the 
l°n&  run  ^  therefore  keeps  the  same  side  always 
towards  the  earth  :  we  see  to-day  precisely  the 
same  face  and  aspect  of  the  moon  as  Galileo  did 
when  he  first  looked  at  it  with  his  telescope,  and 
the  same  will  continue  to  be  the  case  for  thousands 
of  years  more,  if  not  forever. 

U  It  is  difficult  for  some  to  see  why  a  motion  of  this 

FIG.  79.  SC)rt  should  be  considered  a  rotation  of  the  moon,  since 

it  is  essentially  like  the  motion  of  a  ball  carried  on  a 
revolving  crank.  See  Fig.  79.  Such  a  ball,  they  say,  "revolves  around  the 
shaft,  but  does  not  rotate  on  its  own  axis."  It  does  rotate,  however.  The 
shaft  being  vertical  and  the  crank  horizontal,  suppose  that  a  compass 
needle  be  substituted  for  the  ball,  as  in  Fig.  80.  The  pivot  turns  under- 
neath it  as  the  crank  whirls,  but  the  compass 
needle  does  not  rotate,  maintaining  always 
its  own  direction  with  the  marked  end  north. 
On  the  other  hand,  if  we  mark  one  side  of 
the  ball  (in  the  preceding  figure),  we  shall 
find  the  marked  side  presented  successively  to 
every  point  of  the  compass  as  the  crank  re- 
volves, so  that  the  ball  as  really  turns  on  its 
own  axis  as  if  it  were  whirling  upon  a  pin  FIG.  80. 

fastened  to  a  table.     The  ball  has  two  dis- 

tinct motions  by  virtue  of  its  connection  with  the  crank  :  first,  the  motion 
of  translation,  which  carries  its  centre  of  gravity,  like  that  of  the  compass 
needle,  in  a  circle  around  the  axis  of  the  shaft  ;  secondly,  an  additional 
motion  of  rotation  around  a  line  drawn  through  its  centre  of  gravity  parallel 
to  the  shaft. 

248*.  Definition  of  Rotation.  —  A  body  "  rotates  "  whenever  a  line 
drawn  from  its  centre  of  gravity  outward,  through  any  point  selected  at  random 
in  its  mass,  describes  a  circle  in  the  heavens.  In  every  rotating  body,  one  such 
line  can  be  so  drawn  that  the  circle  described  by  it  in  the  sky  becomes  in- 
finitely small.  This  is  the  axis  of  the  body.  Another  set  of  points  can  be 
found  such  that  lines  drawn  from  the  centre  of  gravity  outward  through 


166 


THE   MOON. 


them  describe  a  great  circle  in  the  sky  90°  distant  from  the  point  pierced  by 
the  axis,  and  these  points  constitute  the  equator  of  the  body. 

249.  Librations  of  the  Moon.  —  1.  Libration  in  Latitude.  The 
axis  of  revolution  of  the  moon  is  not  perpendicular  to  its  orbit.  It 
makes  a  constant  angle  of  about  88£°  with  the  ecliptic,  and  the 
moon's  equator  is  so  placed  that  it  is  always  edge-wise  to  the  earth 
when  the  moon  is  at  her  node,  being  maintained  in  that  position  by 
an  action  of  the  earth,  which  produces  a  precessional  motion  of  the 
moon's  axis.  The  angle  between  the  moon's  equator  and  the  plane 
of  her  orbit,  therefore,  is  1-J-0  +  the  inclination  of  the  moon's  orbit, 
which  together  make  up  an  angle  of  a  little  more  than  6^-° ;  but,  as 
the  inclination  of  the  moon's  orbit  to  the  ecliptic  is  constantly  vary- 
ing slightly,  this  inclination  of  the  moon's  axis  to  her  orbit  also 
changes  correspondingly.  This  inclination  of  the  moon's  axis  pro- 
duces changes  in  the  aspect  of  the  moon  towards  the  earth  similar 
to  those  produced  by  the  inclination  of  the  earth's  axis  towards  the 
ecliptic.  At  one  time,  just  as  the  north  pole  of  the  earth  is  turned 
towards  the  sun,  so  also  the  north  pole  of  the  moon  is  tipped  towards 
the  earth  at  an  angle  of  6£°,  and  in  the  opposite  half  of  the  moon's 
orbit  the  south  pole  is  similarly  presented  to  us. 

The  period  of  this  libration  is  the  time  of  the  moon's  revolution  from 

node  to  node,  called  a  nodical 
revolution.  This  is  27.21  days 
—  about  2  hours  and  38  min- 
utes shorter  than  the  sidereal 
revolution  of  the  moon,  since 
the  nodes  always  move  west- 
ward, completing  the  circuit  in 
about  19  years. 

250.  2.  Libration  in  Lon- 
gitude. The  moon's  orbit 
being  eccentric,  she  moves 
faster  when  near  perigee, 
and  slower  when  near  apo- 
gee ;  half-way  between  peri- 
gee and  apogee  she  is  more 
than  6°  ahead  of  the  position 

she  would  have  if  she  had  moved  with  the  mean  angular  velocity. 

Now  the  rotation  is  uniform.1     A  point,  therefore,  on  the  moon's 


FIG.  81.  —  The  Libration  in  Longitude. 


1  Very  nearly  ;  a  minute  "physical  libration "  of  about  3£'  affects  it  slightly. 


ITERATIONS.  167 

surface  which  is  directed  toward  the  earth  at  perigee  will  not 
have  revolved  far  enough  to  keep  it  directed  toward  the  earth  when 
she  is  half-way  (in  time)  between  perigee  and  apogee,  as  is  evident 
from  Fig.  81.  For  in  the  quarter-month  next  following  the  perigee, 
the  moon  will  travel  to  a  point  Jf,  considerably  more  than  half-way  to 
apogee.  But  the  point  a  will  have  made  only  one  quarter-turn,  which 
is  not  enough  to  bring  it  to  the  line  ME.  We  shall  therefore  see  a  little 
around  the  western  edge.  Similarly  on  the  other  side  of  the  orbit, 
half-way  between  apogee  and  perigee,  we  shall  look  around  the  eastern 
edge  to  the  same  extent.  At  perigee  and  apogee  both,  the  libration 
is,  of  course,  zero.  The  amount  of  this  libration  is  evidently  at  any 
moment  just  the  same  as  that  of  the  so-called  "  equation  of  the  centre," 
which,  it  will  be  remembered,  is  the  difference  between  the  mean  and 
true  anomalies  of  the  moon  at  any  moment.  Its  maximum  possible 
value  is  7°45'. 

The  period  of  this  libration  is  the  time  it  takes  the  moon  to  go  around 
from  perigee  to  perigee  —  the  so-called  anomalistic  revolution,  which  is  27.555 
days,  about  5  hours  and  36  minutes  longer  than  the  sidereal  month,  and  8 
hours  14  minutes  longer  than  the  moon's  nodical  revolution,  which  deter- 
mines the  libration  in  latitude. 

The  cause  of  the  increased  length  of  the  anomalistic  revolution  is  of 
course  the  fact  that  the  line  of  apsides  continually  advances  eastward,  mak- 
ing one  revolution  every  nine  years.  (Art.  238.) 

251.  3.  Diurnal  Libration.     This  is  strictly  a  libration  not  of  the 
moon,  but  of  the  observer ;  still,  as  far  as  the  aspect  of    the  moon 
goes,  the  effect  is  precisely  the  same  as  if  it  were  a  true  lunar  libra- 
tion.   The  moon's  motions  have  reference  to  the  earth's  centre.    We, 
on  the  surface  of  the  earth,  look  down  over  the  western  edge  of  the 
moon  when  it  is  rising,  and  over  the  eastern  when  it  is  setting,  by 
an  amount  which  is  equal  to  the  semi-diameter  of  the  earth  as  seen 
from  the  moon  ;  that  is,  about  one  degree  (the  moon's  parallax). 

On  the  whole,  taking  all  three  librations  into  account,  we  see  con- 
siderably more  than  half  the  moon,  the  portion  which  never  disappears 
being  about  forty-one  per  cent  of  the  moon's  surface,  that  never  visi- 
ble also  forty-one  per  cent,  while  that  which  is  alternately  visible  and 
invisible  is  eighteen  per  cent. 

252.  The  agreement  between  the  moon's  time  of  rotation  and  of 
her  orbital  revolution  cannot  be  accidental.     It  is  probably  due  to  the 
action  of  the  earth  on  some  slight  protuberance  on  the  moon's  surface, 


168 


THE   MOON. 


analogous  to  a  tidal  wave.  If  the  moon  were  ever  plastic  the  earth's 
attraction  must  necessarily  have  produced  a  tidal  "bulge"  upon  her 
surface,  and  the  effect  would  ultimately  be  to  force  an  agreement 
between  the  lunar  day  and  the  sidereal  month.  The  subject  will  be 
resumed  later.  (See  Arts.  483-484.) 

253.  The  Phases  of  the  Moon.  —  Since  the  moon  is  an  opaque 
globe,  shining  entirely  by  reflected  light,  we  can  see  only  that  hemi- 
sphere of  her  surface  which  happens  to  be  illuminated,  and  of  course 


FIG.  82.  — Explanation  of  the  Phases  of  the  Moon 


only  that  part  of  the  illuminated  hemisphere  which  is  at  the  time  turned 
towards  the  earth.  At  new  moon,  when  the  moon  is  between  the 
earth  and  the  sun,  the  dark  side  is  towards  us.  A  week  later,  at  the 
end  of  the  first  quarter,  half  of  the  illuminated  hemisphere  is  seen, 
and  we  have  the  half  moon,  just  as  we  do  a  week  after  the  full.  Be- 
tween the  new  moon  and  the  half  moon,  during  the  first  and  last 
quarters  of  the  lunation,  we  see  less  than  half  of  the  illuminated  por- 
tion, and  then  have  the  "  crescent "  phase.  See  Fig.  82  (in  which  the 


THE    MOON'S    PHASES.  169 

light  is  supposed  to  come  from  a  point  far  above  the  moon's  orbit). 
Between  the  half  moon  and  the  full,  during  the  second  and  third 
quarters  of  the  lunation,  we  see  more  than  half  of  the  moon's  illumi- 
nated side,  and  have  what  is  called  the  "  gibbous  "  phase. 

Since  the  terminator  or  line  which  separates  the  dark  portion  of  the 
disc  from  the  bright  is  always  a  semi-ellipse  (being  a  semi-circle  viewed 
obliquely),  the  illuminated  surface  is  always  a  figure  made  up  of  a 
semi-circle  plus  or  minus  a  semi-ellipse,  as  shown  in  Fig.  83,  A. 

It  is  sometimes  incorrectly  attempted  to  represent  the  crescent  form 
by  a  construction  like  Fig.  83,  B  (where  a 
smaller  circle  is  cut  by  a  larger  one).  It  is 
to  be  noticed  that  ab,  the  line  which  joints 
the  cusps,  is  always  perpendicular  to  the  line 
directed  to  the  sun,  and  the  horns  are  always 
turned  away  from  the  sun ;  so  that  the  precise 
position  in  which  they  will  stand  at  any  time  is 
always  predictable,  and  has  nothing  whatsoever 

to  do  with  the  weather.  Artists  are  sometimes  careless  in  the  manner  in 
which  they  introduce  the  moon  into  landscapes.  One  occasionally  sees  the 
moon  near  the  horizon  with  the  horns  turned  downwards,  a  piece  of  drawing 
fit  to  go  with  Hogarth's  barrel  which  shows  both  its  heads  at  once. 

254.  Earth-Shine  on  the  Moon. —  Near  the  time  of  new  moon  the  whole 
disc  of  the  satellite  is  easily  visible,  the  portion  on  which  sunlight  does  not 
fall  being  illuminated  by  a  pale  ruddy  light.     This  light  is  earth-shine,  the 
earth  as  seen  from  the  moon  being  then  nearly  full ;  for  seen  from  the  moon 
the  earth  shows  all  the  phases  that  the  moon  does,  the  earth's  phase  in  every 
case  being  exactly  supplementary  to  that  of  the  moon  as  seen  by  us. 

As  the  earth  has  a  diameter  nearly  four  times  that  of  the  moon,  the  earth- 
shine  at  any  phase  would  be  about  thirteen  times  as  strong  as  moonlight,  if 
the  reflective  power  of  the  earth's  surface  were  the  same.  Probably,  taking 
the  clouds  and  snow  into  account,  the  earth's  surface  on  the  whole  is  rather 
more  brilliant  than  the  moon's,  so  that  near  new  moon  the  earth-shine,  by 
which  the  dark  side  of  the  moon  is  then  illuminated,  is  from  fifteen  to 
twenty  times  as  strong  as  full  moonlight.  The  ruddy  color  is  due  to  the 
fact  that  light  sent  to  the  moon  from  the  earth  has  twice  penetrated  our 
atmosphere  and  so  has  acquired  the  sunset  tinge. 

255.  Physical  Characteristics  of  the  Moon.  —  1.  Its  Atmosphere. 
The  moon's  atmosphere,  if  it  has  any  at  all,  is  extremely  rare,  prob- 
ably not  producing  a  barometric  pressure  to  exceed  ^V  of  an  inch 
of  mercury,  or  ^^  of  the  pressure  at  the  earth's  surface.     The 
evidence  on  this  point  is  twofold. 


170  THE    MOON. 

(a)  The  telescopic  appearance.  The  parts  of  the  moon  near  the 
edge  of  the  disc,  which,  if  there  were  any  atmosphere,  would  be 
seen  through  its  greatest  possible  depth,  are  seen  without  the  least 
distortion :  there  is  no  haze,  and  all  shadows  are  perfectly  black. 
There  is  no  sensible  twilight  at  the  cusps  of  the  moon  ;  no  evidences 
of  clouds  or  storms,  or  anything  like  atmospheric  phenomena. 

(6)  The  absence  of  refraction  when  the  moon  intervenes  between 
us  and  any  more  distant  object.  For  instance,  at  an  eclipse  of  the 
sun  there  is  no  distortion  of  the  sun's  limb  where  the  moon  cuts  it, 
nor  any  ring  of  light  running  out  on  the  edge  of  the  moon  like  that 
which  encircles  the  disc  of  Venus  at  the  time  of  a  transit.  The  most 
striking  evidence  of  this  sort  comes,  however,  from  occupations  of 
the  stars.  When  the_mpon  hides  a  star  from  sight,  the  phenome- 
non, if  it  occurs  at  the  moon's  dark  edge,  is  an  exceedingly  striking 
one.  The  star  retains  its  full  brightness_Jn  th0  fi^ld  of  the  tele- 
^ warning,  it  simply  is  jiot  there, 
the  disappearance  generally  being  n.bsol n tel y  i n atan fo-" pn" «  Its  reap- 
pearance is  of  the  same  sort,  and  still  more  startling.  Now  if  the 
moon  had  any  perceptible  atmosphere  (or  the  star  any  sensible  diam- 
eter) the  disappearance  would  be  gradual.  The  star  would  change 
color,  become  distorted,  and  fade  away  more  or  less  gradually. 

The  spectroscope  adds  its  evidence  in  the  same  direction.  There  is 
no  modification  of  the  spectrum  of  the  star  in  any  respect  at  the  time 
of  its  disappearance  ;  and  we  may  add  that  the  spectrum  of  moonlight 
is  identical  with  that  of  sunlight  pure  and  simple,  there  being  no 
traces  of  any  effect  whatever  produced  upon  the  sunlight  by  its  re- 
flection from  the  moon,  nor  any  signs  of  its  having  passed  through 
an  atmosphere. 

256.  The  time  during  which  a  star  would  be  hidden  behind  the 
moon  would  also  be  decreased  by  the  refraction  of  any  sensible 
atmosphere,  making  the  observed  duration  of  an  occultation  less 
than  that  computed  from  the  known  diameter  of  the  moon  and  its  rate 
of  motion.  Certain  Greenwich  observations  apparently  show  a  differ- 
ence, amounting  to  about  two  seconds  of  time.  This  may  possibly  be 
due  in  some  part  to  the  action  of  a  real,  but  exceedingly  rare,  lunar 
atmosphere  ;  for  if  the  whole  phenomenon  were  due  simply  to  atmos- 
pheric action,  it  would  indicate  an  atmosphere  having  a  density  about 
•J^J-Q-  part  of  our  own,  —  far  within  the  limits  which  were  stated  above. 
But  the  difference  may  be,  and  very  probably  is,  attributable,  in  part 
at  least,  to  a  slight  error  in  the  measured  diameter  of  the  moon,  due  to 


. 

ATMOSPHERE   OF   THE   MOON.  171 

irradiation:  the  diameter  of  a  bright  object  always  appears  a  little 
larger  than  it  really  is.  An  error  of  about  2"  of  this  sort  would 
explain  the  whole  discrepancy,  without  any  need  of  help  from  an 
atmosphere. 

257.  What  has  become  of  the  Moon's  Atmosphere,  —  If  the  moon 
ever  formed  a  part  of  the  same  mass  as  the  earth,  she  must  once  have  had 
an  atmosphere.      There  are  a  number  of  possible  and  more  or  less  probable 
hypotheses  to  account  for  its  disappearance.     It  has  been  surmised  (1)  that 
there  may  be  great  cavities  left  within  the  moon's  mass  by  volcanic  eruptions, 
and  that  the  rocks  themselves  have  been  transformed  into  a  sort  of  pumice- 
stone  structure,  and  that  the  air  has  retired  into  these  internal  cavities. 

(2)  That  the  air  has  been  absorbed  by  the  inner  lunar  rocks  in  cooling. 
A  heated  rock  expels  any  gases  that  it  may  have  absorbed ;  but  if  it  after- 
wards cools  slowly,  it  reabsorbs  them,  and  can  take  up  a  very  great  quantity. 
The  earth's  core  is  supposed  to  be  now  too  intensely  heated  to  absorb  much 
gas  ;  but  if  it  goes  on  cooling,  it  will  absorb  more  and  more,  and  in  time  it 
may  rob  the  surface  of  the  earth  of  all  its  air.  There  are  still  other  hypoth- 
eses,1 which  we  can  not  take  space  even  to  mention. 

258.  Water  on  the  Moon's  Surface.  —  Of  course  without  an  atmos- 
phere there  can  be  no  water,  since  the  water  would  immediately 
evaporate  and  form  an  atmosphere  of  water  vapor  if  there  were  no 
air  present.     It  is  not  impossible,  however,  or  even  improbable,  that 
solid  water,  that  is,  ice  and  snow,  may  exist  on  the  moon's  surface 
at  a  temperature  too  low  for  any  sensible  evaporation.     There  are 
many  things  in  the  moon's  appearance  that  seem  to  indicate  the 
former  existence  of  seas  and  oceans  on  her  surface,  and  the  same 
hypotheses  have  been  suggested  to  account  for  their  disappearance 
that  were  suggested  in  the  case  of  the  moon's  atmosphere.     It  may 
be  added  also  that  many  kinds  of  molten  rock  in  crystallizing  would 
take  up  large  quantities  of  water  of  crystallization,  not  merely  ab- 
sorbed as  a  sponge  absorbs  water,  but  chemically  united  with  the 
other  constituents  of  the  rock.     In  whatever  way,  however,  it  may 
have  come  about,  it  is  certain  that  now  no  substances  that  are  gaseous, 
or  that  can  be  evaporated  at  low  temperatures,  exist  in  any  quantity 
on  the  moon's  surface  —  at  least,  not  on  our  side  of  the  moon. 

There  have  been  speculations  that  on  the  other  side  —  that  celestial  coun- 
try so  near  us  and  so  absolutely  concealed  from  us  —  there  may  be  air  and 
water  and  abundant  life  ;  the  idea  being  that  our  side  of  the  moon  is  a  great 
table-land  many  miles  in  elevation,  while  the  other  side  is  a  corresponding 

1  See  also  note  on  page  181. 


THE   MOON. 

depression,  like  the  valley  of  the  Caspian  Sea,  only  vastly  deeper.  An  in- 
sufficiently grounded  conclusion  of  Hansen's,  that  the  centre  of  gravity  of 
the  moon  is  some  thirty  miles  farther  from  us  than  its  centre  of  figure,  for  a 
time  gave  color  to  the  idea,  but  it  is  now  practically  abandoned,  Hansen's 
conclusion  having  been  shown  to  be  unwarranted  by  the  facts. 

259.  The  Moon's  Light.  —  As  to  quality  it  is  simple  sunlight,  show- 
ing a  spectrum  which,  as  has  been  said,  is  identical  in  every  detail 
with  that  of  light  coming  directly  from  the  sun.     Its  brightness  as 
compared  with  that  of  sunlight  is  difficult  to  measure  accurately,  and 
different  experimenters  have  found  results  for  the  ratio  between  full 
moonlight  and  sunlight  ranging  all  the  way  from  -^^-^  (Bouguer) 
to   -g^oWtf  (Wollaston).     The  value   now  usually  accepted    is   that 
determined  by  Zollner,  viz.,  ^T-§Vo-o-     According  to  this,  if  the  whole . 
visible  hemisphere  were  packed  with  full  moons,  we  should  receive 
from  it  about  one-eighth  part  of  the  light  of  the  sun. 

It  is  found,  also,  that  the  half  moon  does  not  give  even  nearly  half  as 
much  light  as  the  full  moon.  The  law  which  connects  the  phase  of  the 
moon  with  the  amount  of  light  given  at  the  time,  is  rather  complicated,  but 
the  gist  of  the  matter  is  that  at  any  time,  except  at  the  full,  the  visible  sur- 
face is  more  or  less  darkened  by  the  shadows  cast  by  the  irregularities  of  the 
surface.  Zollner  has  calculated  that  an  average  angle  of  52°  for  these  eleva- 
tions and  depressions  would  account  for  the  law  of  illumination  actually 
observed. 

The  average  "  albedo ,"  or  reflecting  power  of  the  moon's  surface, 
Zollner  states  as  0.174;  that  is,  the  moon's  surface  reflects  a  little 
more  than  one-sixth  part  of  the  light  that  falls  upon  it.  This  is  about 
the  albedo  of  a  rather  light-colored  sandstone,  and  agrees  well  with 
the  estimate  of  Sir  John  Herschel,  who  found  the  moon  to  be  very 
exactly  of  the  same  brightness  as  the  rock  of  Table  Mountain  when 
it  was  setting  behind  it,  illuminated  as  were  the  rocks  themselves  by 
the  light  of  the  rising  sun.  There  are,  however,  great  variations  in 
the  brightness  of  different  portions  of  the  moon's  surface.  Some 
spots  are  nearly  as  white  as  snow  or  salt,  and  others  as  dark  as 
slate. 

260.  Heat  of  the  Moon.  —  For  a  long  time  it  was  impossible  to 
detect  the  moon's  heat.     It  is  too  feeble  to  be  detected  by  the  most 
delicate  mercurial  thermometer  even  when  concentrated  by  a  large 
Lens.     The  first  sensible  effect  was  obtained  by  Melloni,  in  1846, 


. 

TEMPER  ATUKE   OF   THE   MOON.  173 

with  the  then  newly  invented  thermopile,  by  a  series  of  observations 
from  the  summit  of  Vesuvius.  Since  then  several  physicists  have 
worked  upon  the  subject  with  more  or  less  success,  especially  Lord 
Rosse  and  Boys  in  Great  Britain,  and  Langley,  Hutchins,  and  Very 
in  the  United  States.  With  modern  apparatus  there  is  no  difficulty 
in  detecting  the  lunar  heat,  but  measurements  are  extremely  difficult 
and  liable  to  error.  A  considerable  percentage  of  the  lunar  heat 
seems  to  be  heat  simply  reflected  (like  light),  while  the  rest,  perhaps 
three-fourths  of  the  whole,  is  "obscure  heat"  ;  that  is,  heat  which 
has  been  first  absorbed  by  the  moon's  surface  and  then  radiated,  like 
the  heat  from  a  brick  surface  that  has  been  warmed  by  sunshine. 
This  is  shown  by  the  fact  that  a  comparatively  thin  plate  of  glass 
cuts  off  some  86  per  cent  of  the  heat  received  from  the  moon  in  the 
same  way  that  it  does  the  heat  of  a  stove,  while  the  heat  of  direct 
sunlight,  or  of  an  electric  arc,  would  pass  through  the  same  plate 
with  very  little  diminution.  The  same  thing  appears  also  from 
direct  measurements  upon  the  heat-spectrum  of  the  moon  made  by 
Langley  with  his  bolometer,  described  further  on.  (Art.  343.) 

The  amount  of  heat  sent  by  the  full  moon  to  the  earth  has  been  estimated 
by  Lord  Rosse  as  TO&TO  of  that  sent  us  by  the  sun  ;  Hutchins'  measures  in 
1888  make  it  only 


261.  As  to  the  temperature  of  the  moon's  surface,  it  is  difficult  to 
affirm  much  with  certainty.  On  one  hand,  the  lunar  rocks  are  ex- 
posed to  the  sun's  rays  in  a  cloudless  sky  for  fourteen  days  at  a 
time,  so  that  if  they  were  blanketed  by  air  like  our  own  rocks  they 
would  certainly  become  intensely  heated.  Some  years  ago,  Lord 
Rosse  inferred  from  his  observations  that  the  temperature  of  the 
lunar  surface  rose  at  its  maximum  (about  three  days  after  full  moon) 
far  above  that  of  boiling  water.1  But  his  own  later  investigations 
and  those  of  Langley  throw  great  doubt  on  this  conclusion.  There 
is  no  air-blanket  at  the  moon's  surface  to  prevent  it  from  losing 
heat;2  and  it  now  seems  rather  more  probable  that  the  temperature 
on  the  equator  at  lunar  noon  rises  very  high,  but  falls  correspond- 
ingly low  as  soon  as  sunlight  is  withdrawn.  So  far  as  we  can 
judge,  the  condition  of  things  on  the  moon's  surface  must  corre- 
spond to  an  elevation  many  times  higher  than  any  mountain  on  the 
earth  ;  for  no  terrestrial  mountain  is  so  high  that  the  density  of 
the  air  at  its  summit  is  even  nearly  as  low  as  that  of  the  densest 
supposable  lunar  atmosphere. 

1  See  note  on  page  183. 

2  See  Art.  377  for  Rosse's  eclipse  observations  bearing  on  this  point. 


174  THE    MOON. 

This  idea  that  the  moon  is  very  cold  is  borne  out,  also,  by  the 
fact  that  the  bolometer  shows  the  presence,  in  the  lunar  radiations, 
of  a  considerable  quantity  of  heat  having  a  wave-length  greater  than 
that  of  the  heat  radiated  from  a  block  of  ice. 

On  the  dark  portion,  during  the  long  f ourteen-days  night  the  tem- 
perature must  probably  fall  at  least  as  far  as  —  200°  F. 

262.  Lunar  Influences  on  the  Earth.  —  The  moon's  attraction  co- 
operates with  that  of  the  sun  in  producing  tides,  of  which  we  shall 
speak  hereafter.     There  are  also  certain  distinctly  ascertained  dis- 
turbances of  terrestrial  magnetism  connected  with  the  approach  and 
recession  of  the  moon  at  perigee  and  apogee  j    and  this  ends  the 
chapter  of  ascertained  lunar  influences. 

The  multitude  of  current  beliefs  as  to  the  controlling  influence  of  the 
moon's  phases  and  changes  over  the  weather  and  the  various  conditions  of 
life  are  mostly  unfounded,  and  in  the  strict  sense  of  the  word  "supersti- 
tions," —  mere  survivals  from  a  past  credulity. 

It  is  quite  certain  that  if  there  is  any  influence  at  all  of  the  sort  it  is  ex- 
tremely slight  —  so  slight  that  it  cannot  be  demonstrated  with  certainty, 
although  numerous  investigations  have  been  made  expressly  for  the  purpose 
of  detecting  it.  We  have  never  been  able  to  ascertain,  for  instance,  with 
certainty,  whether  it  is  warmer  or  not,  or  less  cloudy  or  not,  at  the  time  of  the 
full  moon.  Different  investigations  have  led  to  contradictory  results. 

As  to  the  supposed  connection  between  "  change  of  the  moon  "  and  changes 
of  the  weather,  it  should  be  enough  to  note  that  even  within  the  United 
States  the  weather  changes  are  not  simultaneous  (in  Kansas  and  Maine,  for 
instance),  as  they  should  be  if  they  were  due  to  the  changing  phases  of  the 
moon.  Since,  however,  a  change  of  the  moon  occurs  every  week,  every 
weather  change  must  necessarily  occur  within  about  three  days  and  a  half 
of  a  lunar  change,  and  half  of  them  ought  to  fall  within  about  forty-five 
hours,  even  if  perfectly  independent. 

Now  it  requires  only  a  very  slight  prepossession  in  favor  of  a  belief  in  the 
effectiveness  of  the  moon's  changes  to  make  one  forget  a  few  of  the  weather 
changes  that  occur  too  far  from  the  proper  time.  Coincidences  enough  can 
easily  be  found  to  justify  a  preexisting  belief. 

THE   MOON'S   SURFACE. 

263.  Even  to  the   naked  eye  the  moon  is  a  beautiful 
diversified  with  darker  and  lighter  markings  which  have  given' 
to  numerous  popular  superstitions.  With  a  powerful  telescope  thes 
naked-eye  markings  mostly  vanish,  and  are  replaced  by  a  countless 
multitude  of  smaller  details,  which  are  interesting  in  the  highest 
degree.     The  moon  on  the  whole,  on  account  of  this  diversity  of 

i 


THE    MOOD'S    SURFACE.  175 

detail,  is  the  finest  of  all  telescopic  objects ;  especially  to  moderate- 
sized  instruments,  say  from  six  to  ten  inches  in  diameter,  which 
generally  give  a  more  pleasing  view  of  our  satellite  than  instruments 
either  larger  or  smaller. 

264  How  near  the  Telescope  brings  the  Moon.  —  An  instrument 
of  this  size,  with  magnifying  powers  between  250  and  500,  brings 
up  the  moon  virtually, to  a  distance  ranging  from  1000  miles  to  500 ; 
and  since  an  object  a  mile  in  diameter  on  the  moon  subtends  an 
angle  of  about  0".86,  with  the  higher  powers  of  such  an  instrument 
objects  less  than  a  mile  in  diameter  become  visible  under  favorable 
atmospheric  conditions.  A  long  line  or  streak,  even  less  than  a 
quarter  of  a  mile  across,  could  probably  be  seen.  With  larger  tele- 
scopes the  power  can  now  and  then  be  carried  at  least  twice  as  high, 
and  correspondingly  smaller  details  made  out.  When  everything  is 
at  its  best,  the  great  Lick  telescope  of  36  inches  aperture,  with  a 
power  of  2500  or  so,  may  possibly  reduce  the  virtual  distance  of  our 
satellite  to  about  100  miles  for  visual  purposes.  It  is  evident  that 
while  with  our  telescopes  we  should  be  able  to  see  such  objects  as 
lakes,  rivers,  forests,  and  great  cities,  if  they  exist  on  the  moon,  it 
will  be  hopeless  to  expect  to  distinguish  single  buildings,  or  any  of 
the  ordinary  operations  and  indications  of  life,  if  such  there  are. 

There  are  a  few  mountains  on  the  earth  from  which  a  range  of  100  miles 
is  obtained  in  the  landscape.  Those  who  have  seen  such  a  landscape  know 
how  little  is  to  be  made  out  with  the  naked  eye  at  that  distance.  Still,  the 
comparison  is  not  quite  fair,  because  in  looking  at  a  terrestrial  object  a  hun- 
dred miles  away  the  line  of  vision  passes  through  a  dense  atmosphere,  while 
in  looking  upward  towards  the  moon  it  penetrates  a  much  less  thickness 
of  air. 

265.  The  Moon's  Surface  Structure.  — The  moon's  surface  for  the 
most  part  is  extremely  uneven  and  broken,  far  more  so  than  that  of 
the  earth.  The  structure,  however,  is  not  like  that  of  the  earth's 
surface.  On  the  earth  the  mountains  are  mostly  in  long  ranges,  such 
as  the  Alps,  the  Andes,  and  Himalayas.  On  the  moon  such  moun- 
tain ranges  are  few  in  number,  though  they  exist ;  but  the  surface  is 
pitted  all  over  with  great  craters,  resembling  very  closely  the  vol- 
canic craters  on  the  earth's  surface,  though  on  an  immensely  greater 
scale.  One  of  the  largest  craters  upon  the  earth,  if  not  the  largest, 
is  the  Aso  San  in  Japan,  about  seven  miles  across.  Many  of  those 
on  the  moon  are  fifty  and  sixty  miles  in  diameter,  and  some  are 


176  THE   MOON. 

over  100  miles  across,  while  smaller  ones  from  a  half-mile  to  eight 
or  ten  miles  in  diameter  are  counted  by  the  thousand. 

The  normal  lunar  crater  is  nearly  circular,  surrounded  by  an  ele- 
vated ring  of  mountains  which  rise  anywhere  from  1000  to  20000 

feet  above  the  surround- 
ing country.  Within  the 
floor  of  the  crater  the 
surface  may  be  either 
above  or  below  the  out- 
side level.  Some  craters 
are  deep,  some  filled 
nearly  to  the  brim.  In 
some  cases  the  surround- 
ing mountain  ring  is  en- 

Fio.  84.  -A  Normal  Lunar  Crater  (Nasmyth).  tirel7     absent>     and     the 

crater  is  a  mere  hole  in 
the  plain.  In  the  centre  of  the  crater  there  usually  rises  a  group 
of  peaks,  of  about  the  same  elevation  as  the  encircling  ring,  and 
these  peaks  often  show  holes  or  craterlets  in  their  summits. 

In  most  cases  the  resemblance  of  these  formations  to  terrestrial  volcanic 
structures,  like  those  exemplified  by  Vesuvius  and  others  in  the  surround- 
ing region,  makes  it  natural  to  assume  that  they  had  a  similar  origin. 
This,  however,  is  not  absolutely  certain,  for  there  are  considerable  difficul- 
ties in  the  way,  especially  in  the  case  of  the  great  "Bulwark  Plains,"  so- 
called,  which  are  so  extensive  that  a  person  standing  in  the  centre  could  not 
see  the  summit  of  the  surrounding  ring  at  any  point ;  and  yet  no  line  of 
demarcation  can  be  drawn  between  them  and  the  smaller  craters.  The  series 
is  continuous.  Moreover,  on  the  earth,  volcanoes  necessarily  require  the 
action  of  air  and  water,  which  do  not  now  exist  on  the  moon.  It  is  obvious, 
therefore,  that  if  these  lunar  craters  are  the  result  of  true  volcanic  eruptions, 
they  must  be  'fossil'  formations  ;  for  it  is  quite  certain 1  that  no  evidence  of 
existing  volcanic  activity  has  ever  been  found.  The  moon's  surface  appears 
to  be  absolutely  quiescent  —  still  in  death. 

On  some  portions  of  the  moon  these  craters  stand  very  thickly. 
Older  craters  have  been  encroached  upon,  or  more  or  less  completely 
obliterated  by  the  newer,  and  the  whole  surface  is  a  chaos,  of  which 
the  counterpart  is  hardly  to  be  found  on  the  earth,  even  in  the 
roughest  portions  of  the  Alps.  This  is  especially  the  case  near  the 
moon's  south  pole.  It  is  noticeable  that,  as  on  the  earth  the  newest 
mountains  are  generally  the  highest,  so  on  the  moon  the  more  newly 
formed  craters  are  generally  deeper  and  more  precipitous  than  the 
-older  ones. 

1  See  note  on  page  183. 


LUNAR    NOMENCLATURE. 


177 


266.  Lunar  Nomenclature.  —  The  great  plains  were  called  by 
Galileo  oceans  or  seas  (Maria),  and  some  of  the  smaller  ones 
marshes  (Paludes)  and  lakes,  for  he  supposed  that  the  grayish  sur- 
faces visible  to  the  naked  eye,  and  conspicuous  in  a  small  telescope, 
were  covered  with  water.  Thus  we  have  the  "  Oceanus  Procellarum," 
the  "  Mare  Imbrium,"  and  a  number  of  other  "  seas/7  of  which 


FIG.  85.  —  Map  of  the  Moon.    (Reduced  from  Neison.) 

"Mare  Fecunditatis,"  "Mare  Serenitatis,"  and  "Mare  Tranquilita- 
tis,"  are  the  most  conspicuous.  There  are  twelve  of  them  in  all, 
and  eight  or  nine  Paludes,  Lacus,  and  Sinus. 

The  ten  mountain  ranges  on  the  moon  are  mostly  named  after 
terrestrial  mountains,  as  Caucasus,  Alpes,  Apennines,  though  two 
or  three  bear  the  names  of  astronomers,  like  Leibnitz,  Dorfel,  etc. 


178 


THE   MOON. 


The  conspicuous  craters  bear  the  names  of  the  more  eminent  ancient 
and  mediaeval  astronomers  and  philosophers,  as  Plato,  Archimedes, 
Tycho,  Copernicus,  Kepler,  and  Gassendi ;  while  hundreds  of  smaller 
and  less  conspicuous  formations  bear  the  names  of  more  modern  or 
less  noted  astronomers. 

The  system  seems  to  have  originated  with  Riccioli  in  1650,  but  most  of 
the  names  have  been  more  recently  assigned  by  the  later  map-makers,  the 
most  eminent  of  whom  have  been  the  German  astronomers  Beer  and  Maed- 
ler  (who  published  their  map  in  1837),  and  Schmidt  of  Athens,  whose  great 
map  of  the  moon,  on  a  scale  seven  feet  in  diameter,  was  published  by  the 
Prussian  government  some  years  ago.  It  is  not  at  all  too  much  to  say  that 
our  maps  of  the  earth's  surface  do  not,  on  the  whole,  compare  in  fulness  and 
accuracy  with  our  maps  of  the  moon.  Of  course  this  is  not  true  of  such 
countries  as  France  and  England,  or  others  that  have  been  trigonometrically 
surveyed ;  but  there  are  no  such  lacunce  in  our  maps  of  the  moon  as  exist  in 
our  maps  of  Asia  and  Africa,  for  instance. 

267.     Other  Lunar  Formations. — The  craters  and  mountains  are 

not  the  only  interesting  for-     

mations  on  the  moon's  sur- 
face. There  are  many  deep, 
narrow,  crooked  valleys  that 
goby  the  name  of  "rills" 
(German  Kitten),  some  of 
which  may  once  have  been 
watercourses.  Then  there 
are  numerous  "  clefts,"  half 
a  mile  or  so  wide  and  of  un- 
known depth,  running  in 
some  cases  several  hundred 
miles,  straight  through  moun- 
tain and  valley,  without  any 
apparent  regard  for  the  ac- 
cidents of  the  surface. 
They  seem  to  be  deep 
cracks  in  the  crust  of  our 
satellite.  Several  of  them 
are  shown  in  Fig.  86.  Most 
curious  and  interesting  of  all  are  the  light-colored  streaks  or  "  rays " 
which  radiate  from  certain  of  the  craters,  extending  in  some  cases 
a  distance  of  several  hundred  miles.  They  are  usually  from  five  to 
ten  miles  wide,  and  neither  elevated  nor  depressed  to  any  extent  with 


FIG.  86.  —  Archimedes  and  the  Apennines  (Nasmyth). 


CHANGES    ON   THE   MOON. 


179 


reference  to  the  general  surface.  They  pass  across  mountain  and 
valley,  and  sometimes  through  craters  without  any  change  in  width 
or  color.  We  do  not  know  whether  they  are  like  the  so-called  "  trap- 
d3'kes  "  on  the  earth,  —  fissures  which  have  been  filled  up  from  below 
with  some  light-colored  material,  —  or  whether  they  are  mere  sur- 
face markings.  No  satisfactory  explanation  has  ever  been  given. 

The  most  remarkable  system  of  "rays"  of  this  kind  is  the  one 
connected  with  the  great  crater  Tycho,  not  very  far  from  the  moon's 
south  pole.  They  are  not  ver}^  conspicuous  until  within  a  few  days 
of  full  moon,  but  at  that  time  they,  and  the  crater  from  which  they 
radiate,  constitute  by  far  the  most  striking  feature  of  the  whole  lunar 
landscape. 

268.  Changes  on  the  Moon.  —  It  is  certain  that  there  are  no  con- 
spicuous changes.  The  ob- 
server has  before  him  no 
such  ever  -  varying  vision 
as  he  would  have  in  look- 
ing toward  the  earth, — 
no  flying  clouds,  no  alter- 
nations of  seasons  with  the 
transformation  of  the  snowy 
wastes  to  green  fields,  nor 
any  considerable  apparent 
movement  of  objects  on  the 
disc.  The  sun  rises  on  them 
slowly  as  they  come  one 
after  the  other  to  the  ter- 
minator, and  sets  as  slowly. 
At  the  same  time  it  is  con- 
fidenth'  maintained  by  many 
observers  that  here  and  there 
changes  are  still  going  on  in 
the  details  of  the  surface. 
Others  as  stoutly  dispute  it. 


FIG.  87.  —  Gassendi  (Nasmyth). 


Probably  the  most  notable  and  best  advocated  instance  of  such  a 
change  is  that  of  the  little  crater  Linne,  in  the  Mare  Serenitatis.  It  was  ob- 
served by  Schroeter  very  early  in  the  century,  and  is  figured  and  described  by 
Beer  and  Maedler  as  being  about  five  and  a  half  or  six  miles  in  diameter,  quite 
deep  and  very  bright.  In  1866  Schmidt,  who  had  several  times  observed  it 
before,  announced  that  it  had  disappeared.  A  few  months  later  it  was 


180  THE   MOON. 

visible  again,  and  there  were  many  reported  changes  in  its  appearance 
during  the  next  year  or  two.  There  is  no  question  that  it  does  not  now  at 
all  agree  in  conspicuousness  and  size  with  the  representation  of  Beer  and 
Maedler,  for  it  is  at  present,  and  has  been  for  several  years,  only  a  minute 
dark  spot,  with  a  whitish  spot  surrounding  it.  Astronomers  would  feel 
more  confident  that  this  was  a  case  of  real  change  were  it  not  that  Schroe- 
ter's  earlier  picture  much  more  resembles  the  present  appearance  than  does 
that  of  Beer  and  Maedler.  As  the  latter  observers  worked  with  rather  a 
small  telescope,  and  had  no  reason  for  taking  any  special  pains  in  the  delin- 
eation of  this  particular  object,  the  evi- 
dence is  less  conclusive  than  it  might 
seem  at  first.  The  change,  however,  if 
real,  was  certainly  as  great  as  in  the 
instance  of  Krakatoa,  the  great  volcano 
whose  eruption  in  1883  filled  the  earth's 
atmosphere  with  smoke  and  vapor  for 
more  than  two  years,  and  caused  the 
"  twilight  conflagrations  "  of  the  sky. 
The  phenomenon  in  the  case  of  Linn6, 
if  real,  was  probably  a  falling  in  of  the 
walls  of  the  crater,  exposing  fresh  un- 
weathered  surfaces. 
FlG  gg  The  difficulty  in  establishing  the 

reality  of  such  changes  lies  mainly  in 

the  great,  but  purely  apparent,  discrepancies  due  to  varying  illumination 
and  to  the  "personal  equation"  of  observers  and  their  telescopes.  Com- 
parisons can  be  safely  instituted  only  between  observations  made  under 
conditions  (lunar,  atmospheric,  instrumental,  and  personal)  which  are  sen- 
sibly identical,  and  such  identity  is  of  course  not  easy  to  secure. 
The  final  appeal  will  be  to  photography.1 

270.  Measurement  of  Lunar  Mountains.  —  The  height  of  a  lunar 
mountain  is  usually  determined  by  measuring  with  a  micrometer,  as 
accurately  as  possible,  the  apparent  length  of  its  shadow,  and  also, 
a  little  more  roughly,  measuring  at  the  same  time  the  distance  of 
the  object  from  the  terminator  and  from  what  may  be  called  the 
"equator  of  illumination,"  —  the  line  6  (in  Fig.  88)  which  bisects 
the  phase  symmetrically.  With  these  data  and  those  supplied  by 
the  almanac,  the  result  is  easily  calculated  by  formulae  given  in 
Neison's  "  Moon."  If  the  mountain  is- favorably  situated,  its  height 
can  be  determined  in  this  way  with  an  error  not  exceeding  five  or 
six  hundred  feet. 

In  some  cases  the  height  is  computed  from  measurements  of  the 

1  See  note  on  page  183. 


MEASUREMENT    OF    LUNAR   MOUNTAINS.  181 

distance  between  the  terminator  and  the  top  of  the  mountain  when 
it  first  catches  the  sunlight,  and  looks  like  a  star  outside  the  termi- 
nator, as  shown  in  Figs.  87  and  88.  But  this  is  less  accurate. 

Many  of  the  lunar  mountains  reach  an  elevation  of  15000  feet 
and  upwards.  One  of  the  highest,  so  situated  that  its  height  can 
be  fairly  measured,  is  Mt.  Huyghens  in  the  range  of  the  'Appenines 
on  the  western  edge  of  the  Mare  Imbrium,  —  a  little  over  18000 
feet  high.  Very  near  the  south  pole,  and  only  visible  in  outline 
under  favorable  conditions  of  libration,  are  the  great  Leibnitz  and 
Doerfel  ranges,  which  are  much  higher,  — .probably  between  25000 
and  30000  feet. 

271.  The  Best  Time  to  Look  at  the  Moon  with  a  Telescope.  —  The 

moon  when  full  is  not  so  satisfactory  an- object  as  when  near  the  half,  be- 
cause at  the  full  moon  there  are  no  shadows,  so  that  at  that  time  the  "relief" 
of  the  surface  structure  is  entirely  lost.  Certain  features,  however,  as  has 
been  before  mentioned,  are  then  best  seen,  as,  for  instance,  the  streaks  or 
rays.  Generally,  any  particular  mountain,  crater,  rill,  or  cleft  is  best  studied 
when  it  is  just  on  or  very  near  the  terminator,  that  is,  at  the  time  when  the 
sun  is  rising  or  setting  near  it,  because  then  the  shadows  are  longest.  The 
best  general  view  of  the  moon  is  that  obtained  a  few  days  after  the  half 
moon,  when  Copernicus  and  Tycho  are  both  near  the  terminator,  and  Plato 
is  still  near  enough  to  it  to  show  very  well. 

272.  Photographs  of  the  Moon A  great  deal  of  attention  has  been 

paid  to  this  subject,  and  some  fine  results  have  been  reached.     The  earliest 
success  was  that  of  Bond  in  1850,  with  the  old  daguerreotype  process ;  then 
followed  the  work  of  De  la  Rue  in  England,  and  of  Dr.  Henry  Draper,  and 
especially  of  Mr.  Rutherfurd  in  this  country.     Rutherfurd's  pictures  have 
remained  absolutely  unrivalled  until  very  recently. 

Since  1885,  however,  great  progress  has  been  made.  In  this  country  the 
Harvard,  the  Lick,  and  the  Yerkes  observatories  have  reached  admirable 
results,  and  in  Europe  the  Paris  observatory.  From  negatives  made  at  the 
Lick  and  Paris  observatories,  and,  more  recently,  by  W.  H.  Pickering  in 
Jamaica,  complete  atlases  of  the  moon  have  been  made.  The  Paris  atlas 
is  especially  fine,  showing  the  features  on  various  scales  corresponding  to 
lunar  diameters  ranging  from  four  to  nine  feet.  But  the  photograph  cannot 
yet  rival  the  eye  in  the  study  of  delicate  details. 

272*.  Note  to  Art.  257.  —  It  is  not  improbable  that  the  extent  and  to 
a  certain  degree  the  composition  of  the  atmosphere  of  a  heavenly  body  may 
depend  directly  upon  its  mass  and  density.  Indeed,  if  the  "  Kinetic  theory  " 
of  gases  is  true,  it  must  necessarily  be  so,  as  was  pointed  out  some  years 
ago  by  Johnstone  Stoney,  of  Dublin.  According  to  this  theory  the  mole- 
cules of  a  gas  are  continually  flying  in  all  directions  with  a  velocity  depend- 


182  THE   MOON. 

ent  upon  their  mass  and  temperature.  Individual  molecules  move,  some 
faster  and  some  slower,  and  a  certain  small  percentage  may  attain  a  speed 
six  or  seven  times  as  great  as  this  mean  velocity.  At  zero  (Cent.)  the 
maximum  molecular  velocity  of  oxygen  is  computed  as  about  1.8  miles  a 
second  ;  that  of  nitrogen,  2.0  ;  that  of  water-vapor,  2.5  ;  that  of  helium,  5.2; 
and  that  of  hydrogen,  7.4, —  values  which  increase  or  decrease  with  the 
temperature. 

Again,  at  any  given  distance  from  a  body  there  is  a  so-called  "critical," 
or  "parabolic,"  velocity  (Arts.  429  and  435)  depending  on  the  mass  of  the 
body ;  and  if  a  particle  at  this  distance  has  a  speed  greater  than  this  para- 
bolic velocity,  it  cannot  be  retained  by  the  body's  gravitational  attraction, 
but  will  fly  off  into  space.  At  the  surface  of  the  sun  this  critical  velocity  is 
about  383  miles  a  second ;  at  the  surface  of  the  earth  it  is  a  little  less  than 
7  ;  and  on  the  moon's  surface  it  is  only  1.5.  It  is  clear,  therefore,  that  if  the 
sun  were  cool,  not  a  molecule  of  any  of  the  gases  we  have  mentioned  above 
would  ever  escape  from  its  atmosphere.  On  the  earth,  however,  hydrogen 
cannot  be  retained  free,  but  only  in  chemical  combination ;  helium  would 
be  likely  to  go  also,  since  a  slight  elevation  of  temperature  above  the  freez- 
ing point  might  increase  its  molecular  velocity  beyond  the  7-mile  limit. 
Oxygen,  nitrogen,  and  water-vapor,  on  the  other  hand,  stay  by  us. 

But  on  the  moon  the  force  of  gravity  is  so  small  that  even  if  she  were 
now  by  some  means  once  more  reheated  and  reclothed  with  an  atmosphere 
like  our  own,  its  molecules  would  one  after  another  take  flight,  and  soon 
leave  her  airless  again. 

This,  however,  all  hangs  upon  the  truth  of  the  kinetic  theory  of  gases, 
which,  though  very  probable,  can  hardly  be  considered  as  yet  completely 
proved. 

It  is  worth  noting,  also,  that  on  this  hypothesis  interplanetary  space  must 
be  populous  with  wandering  molecules  of  the  various  gases,  which,  how- 
ever, no  longer  behave  like  "  gas,"  as  we  know  it  on  the  earth,  because  they 
are  too  far  apart  and  collide  with  each  other  too  seldom  to  enable  them  to 
manifest  the  familiar  gaseous  properties.  Now  these  wandering  molecules 
must  be  continually  entering  a  planet's  atmosphere,  and  when  as  many  arrive 
in  a  day  as  fly  oif  in  the  same  time,  this  atmosphere  ceases  either  to  grow 
or  to  diminish. 


EXERCISES  ON  CHAPTER  VII. 

1.  If  the  moon's  sidereal  period  were  sixty  days,  what  would  be  her  syn- 
odic period?  Ans.    71.7932  days. 

2.  In  that  case,  what  would  be  the  mean  interval  between  her  meridian 
transits?     (See  Art.  235.)  Ans.   24h  20.34m. 

3.  Does  the  moon  rise  every  day? 


EXERCISES.  183 

4  If  the  moon  rises  at  llh  45m  P.M.  on  Wednesday,  when  (approxi- 
mately) will  she  rise  next  V 

5.  What  is  the  lowest  latitude  on  the  earth  where  the  moon  can  remain 
above  the  horizon  for  48  consecutive  hours  ? 

Am.   90°-  (23°  27'  +  5°  8')  =  61° 25'. 

6.  At  what  time  of  the  year  does  the  full  moon  remain  longest  above 
the  horizon  ? 

7.  How  many  times  does  the  moon  turn  on  its  axis  in  a  year? 

8.  Does  the  earth  rise  and  set  for  an  observer  on  the  moon? 

9.  What  determines  the  direction  of  the  horns  of  the  crescent  moon? 

10.  Can  a  star  ever  be  seen  between  the  horns  of  the  moon  ? 

11.  What  point  describes  "  the  orbit  of  the  earth  "  around  the  sun  ? 

Ans.    The  centre  of  gravity  of  the  earth  and  moon. 

12.  Does  the  centre  of  the  sun,  as  seen  from  the  centre  of  the  earth,  fol- 
low the  ecliptic  exactly,  and  if  not,  how  far  can  it  depart  from  it  on  account 
of  the  moon's  action  ?     (See  Arts.  233  and  243.) 

64 
Ans.    The  deviation  may  be  —  x  8".80  x  sin  5°  20'  =  0".6. 

88 

There  are  also  perturbations  of  the  earth  by  the  planets,  producing  ad- 
ditional deviations  of  about  0".5,  so  that  at  times  the  latitude  of  the  sun's 
centre  may  slightly  exceed  ±  I'M. 

NOTE  TO  ART.  261. 

An  investigation  by  Professor  Very,  published  in  January,  1899,  bears  strongly 
in  the  same  direction  as  Lord  Rosse's  result,  indicating  a  maximum  temperature 
on  the  moon's  illuminated  surface  approaching  that  of  boiling  water,  but  falling 
immediately  on  the  withdrawal  of  sunlight. 

NOTE  TO  ARTS.  265  AND  269. 

Professor  W.  H.  Pickering  considers  that  his  observations  and  photographs 
of  1902-1903  show  changes  upon  the  moon's  surface  which  indicate  the  deposi- 
tion of  snow  or  hoar-frost  at  certain  points  during  the  lunar  night,  and  its  sub- 
sequent disappearance  when  the  sun's  rays  reach  it  :  very  much  as  if  a  subdued 
volcanic  activity  still  persisted  in  the  moon  with  vapors  issuing  from  beneath 
through  fissures  and  fumaroles,  as  in  certain  parts  of  the  earth,  Iceland  for 
instance.  But  as  yet  his  views  do  not  seem  to  have  gained  general  acceptance. 


184  THE   SUN. 


CHAPTER   VIIL 

THE   SUN  :     DISTANCE  AND  DIMENSIONS.  —  MASS  AND  DENSITY. 

ROTATION.  —  STUDY  OF  THE  SURFACE  :     GENERAL  VIEWS 

AS  TO  THE  SUN'S   CONSTITUTION.  —  SUN  SPOTS  :     THEIR  AP- 
PEARANCE, NATURE,  DISTRIBUTION,  AND  PERIODICITY. 

273.  THE  SUN  is  simply  a  star;  a  hot,  self-luminous  globe  of 
enormous  magnitude  as  compared  with,  the  earth  and   the  moon, 
though  probably  only  of  medium  size  among  its  stellar  compeers. 
But  to  the  earth  and  the  other  planets  which  circle  around  it,  it  is 
the  grandest  of    all  physical  objects.       Its  attraction  confines  its 
planets  to  their  orbits  and  controls  their  motions,   and   its   rays 
supply  the    energy  which  maintains  every  form  of  activity  upon 
their  surfaces  and  makes  them  habitable. 

274.  Its  Distance  and  Dimensions.  —  Its  distance  may  be  deter- 
mined from  its  horizontal  parallax,  which  is  the  apparent  angular 
semi-diameter  of  the  earth  as  seen  from  the  sun.     The  mean  value 
of  this  parallax  is  probably  very  near  S".S.1 

We  reserve  to  a  separate  chapter  the  discussion  of  the  methods 
by  which  this  most  fundamental  and  important  of  all  astronomical 
data  has  been  ascertained,  merely  remarking  here  that  the  problem 
is  one  of  extreme  practical  difficulty,  though  the  principles  involved 
are  simple  enough. 

Assuming  the  parallax  at  8".8,  the  mean  distance  of  the  sun  (put- 
ting r  for  the  earth's  radius)  equals 

r  -r-  sin  8".8=  23439  X  r. 
With  Clarke's  value  of  r  (Art.  145),  this  gives  149  500000  kilometers, 


1  In  the  American  Ephemeris  the  value  deduced  by  Newcomb  in  1867  was 
used,  viz.,  8". 85.  The  British  "Nautical  Almanack"  used  the  same  value,  and 
the  French  the  value  deduced  by  Leverrier  a  little  earlier,  8".86;  but  more  re- 
cent observations  show  that  the  number  stated,  8 ".8,  is  more  nearly  correct, 
and  since  1000  it  has  been  used  in  all  three  ephemerides. 


THE    SUN'S    DIAMETER. 


185 


or  92  897000  miles ;  which,  however,  is  uncertain  by  at  least  50000 
miles,  and  is  variable,  also,  to  the  extent  of  about  three  million  miles 
on  account  of  the  eccentricity  of  the  earth's  orbit,  the  earth  being 
nearer  the  sun  in  December  than  in  June. 

275.  This  distance  is  so  much  greater  than  any  with  which  we 
have  to  do  on  the  earth  that  it  is  possible  to  reach  a  conception  of 
it  only  by  illustrations  of  some  sort.  Perhaps  the  simplest  is  that 
drawn  from  the  motion  of  a  railway  train.  Such  a  train  going  1000 
miles  a  day  (nearly  forty-two  miles  an  hour)  would  take  254-J-  years 
to  make  the  journey. 

If  sound  were  transmitted  through  interplanetary  space,  and  at 
the  same  rate  as  through  our  own  atmosphere,  it  would  make  the 
passage  in  about  fourteen  years  ;  i.e.,  an  explosion  on  the  sun  would 
be  heard  by  us  fourteen  years  after  it  actually  occurred.  A  cannon- 
ball  moving  unretarded,  at  the  rate  of  1700  feet  per  second,  would 
travel  the  distance  in  nine  years.  Light  does  it  in  499  seconds. 


FIG.  90.  —Dimensions  of  the  Sun  Compared  with  the  Moon's  Orbit. 

276.  Diameter.  —  The  sun's  mean  apparent  diameter  is  32'  04"  ± 
2".  Since  at  the  sun  one  second  equals  450.38  *  miles,  its  diameter 
equals  866500  miles,  or  109J-  times  the  diameter  of  the  earth.  It 
is  quite  possible  that  this  diameter  is  variable  to  the  extent  of  a  few 

1  92  897000  -r  206264.8  =  450.38. 


186  THE  SUN. 

hundred  miles,  since,  as  will  appear  hereafter,  the  sun  (at  least  the 
surface  which  we  see)  is  not  solid. 

Representing  the  sun  by  a  globe  two  feet  in  diameter,  the  earth 
would  be  32¥%  of  an  inch  in  diameter,  —  the  size  of  a  very  small  pea, 
or  a  "  22- calibre  "  round  pellet.  Its  distance  from  the  sun  on  that 
scale  would  be  just  about  220  feet,  and  the  nearest  star  (still  on  the 
same  scale)  would  be  eight  thousand  miles  away,  at  the  antipodes. 

If  we  were  to  place  the  earth  in  the  centre  of  the  sun,  supposing 
it  to  be  hollowed  out,  the  sun's  surface  would  be  433,000  miles  away 
from  us.  Since  the  distance  of  the  moon  is  only  about  239,000 
miles,  it  would  be  only  a  little  more  than  half-way  out  from  the 
earth  to  the  inner  surface  of  the  hollow  globe,  which  would  thus 
form  a  very  good  background  for  the  study  of  the  lunar  motions. 

It  is  perhaps  worth  noticing,  as  a  help  to  memory,  that  the  sun's  diameter 
exceeds  the  earth's  just  about  as  many  times  as  it  is  itself  exceeded  by  the 
radius  of  the  earth's  orbit ;  or,  in  other  words,  the  sun's  diameter  is  nearly  a 
mean  proportional  between  the  earth's  distance  from  the  sun  and  the  earth's 
diameter,  110  being  the  common  ratio. 

277.  Surface  and  Volume.  —  Since  the  surfaces  of  globes  are  pro- 
portional to  the  squares  of  their  radii,  the  surface  of  the  sun  exceeds 
that  of  the  earth  in  the  ratio  of  (109. 5)2  to  1  ;  that  is,  its  surface  is 
about  12,000  times  the  surface  of  the  earth. 

The  volumes  of  spheres  are  proportional  to  the  cubes  of  their  radii ; 
hence  the  sun's  volume  is  (109. 5)3,  or  1,300000  times  that  of  the  earth. 

278.  The  Sun's  Mass.  — The  mass  of  the  sun  is  very  nearly  three 
hundred  and  thirty-two  thousand  times  that  of  the  earth,  subject  to  a 
probable  error  of  at  least  one  per  cent.     There  are  various  ways  of 
getting  at  this  result.     For  our  purpose  here,  perhaps  the  most  con- 
venient is  by  comparing  the  earth's  attraction  for  bodies  at  her  surface 
(as  determined  by  pendulum  experiments)  with  the  attraction  of  the 
sun  for  the  earth,  —  the  central  force  which  keeps  her  in  her  orbit. 
Put/ for  this  force  (measured,  like  gravity,  by  the  velocity  it  gener- 
ates  in  one  second),  g  for  the  force  of  gravity  (32  feet  2  inches 
per  second),  r  the  earth's  radius,  R  the  sun's  distance,  and  let  E 
and  S  be  the  masses  of  the  earth  and  sun  respectively.     Then,  by 
the  law  of  gravitation,  we  have  the  proportion 

*•••*•  f-  -— ©(?/• 

Now,  -  =  23,440  (nearly) . 


THE    SUN'S   MASS.  187 

Its  square  equals  549,433,600.     g  =  386  inches.     To  find  /  we  have 
from  Mechanics  (Physics,  pp.  17  and  28), 


this  being  the  expression  for  the  "  central  force"  in  the  case  of  a 
body  revolving  in  a  circle.  (We  may  neglect  the  eccentricity  of  the 
earth's  orbit  in  a  merely  approximate  treatment  of  the  problem.) 
V  is  the  orbital  velocity  of  the  earth,  which  is  found  by  dividing  the 
circumference  of  the  orbit,  2  TT^,  by  T7,  the  number  of  seconds  in  a 
sidereal  year.  This  velocity  comes  out  18.495  miles  per  second. 
Putting  this  into  formula  (6),  we  get/=  0.2333  inches, 

so  that  £  =  0.0006044  =  —  —  (nearly)  ; 

g  1654 

whence  S  =  Ex  —  —  X  549,433,600  ;  or  S  equals  332,000. 

1654 

We  may  note  in  passing  that  half  of  /  expresses  the  distance  by 
which  the  earth  falls  towards  the  sun  every  second,  just  as  half  g  is 
the  distance  a  body  at  the  earth's  surface  falls  in  a  second.  This 
quantity  (0.116  inch),  a  trifle  more  than  a  ninth  of  an  inch,  is  the 
amount  by  which  the  earth's  orbit  deviates  from  a  straight  line  in  a 
second.  In  travelling  eighteen  and  one-half  miles  the  deflection  is 
only  one-ninth  of  an  inch. 


278*.    By  substituting  for  V  in  equation  (ft),  we  get 


/= 

J.  - 

and  putting  this  value  of /into  equation  (a)  and  reducing,  we  obtain 

S=I7|(  —  W_U- J  J,  (c) 

or,  since  -  =  — — 

(p  being  the  sun's  horizontal  parallax),  we  have  finally 

00 


It  will  be  noticed  that  in  this  expression  the  cube  of  the  parallax  appears, 
and  this  is  the  reason  why  an  uncertainty  of  one  per  cent  in  p  involves  an 
uncertainty  of  three  per  cent  in  S. 


188  THE   SUN. 

In  obtaining  the  mass  of  the  sun  it  will  be  seen  that  we  require 
as  data,  T,  the  length  of  the  sidereal  year  in  seconds  ;  the  value  of 
gravity,  g  (which  is  derived  from  pendulum  experiments) ;  the  radius 
of  the  earth,  r  (deduced  from  geodetic  surveys);  and  finally  (and 
most  difficult  to  get),  the  sun's  parallax,  p,  or  else  the  sun's  distance, 

It ;    giving  us  in  either  case  the  ratio  — • 

279.  The  Sun's  Density.  —  This  density1  as  compared  with  that 
of  the  earth  is  found  by  simply  dividing  its  mass  by  its  volume 
(both  as  compared  with  the  earth) ;  that  is,  it  equals  the  fraction 

332000 


1  300000 


=  0.255, 


a  little  more  than  a  quarter  of  the  earth's  density.  To  get  its  "spe- 
cific gravity"  (i.e.,  density  as  compared  with  water),  we  must  multiply 
this  by  5.58,  the  earth's  mean  specific  gravity.  This  gives  1.41  ; 
that  is,  the  sun's  mean  density  is  not  \%  times  that  of  water,  —  a  most 
significant  result  as  bearing  on  its  physical  condition. 

280.    Superficial  Gravity.  —  This  is  found  by  dividing  its  mass  by 
the  square  of  its  radius  ;  that  is, 

332000 


which  equals  27.6.  A  body  weighing  one  pound  on  the  earth's  sur- 
face would  there  weigh  27.6  pounds.  A  body  would  fall  444  feet 
in  a  second,  instead  of  16  feet,  as  here. 

281.  The  Sun's  Rotation.  —  The  sun's  surface  often  shows  spots 
upon  it,  which  pass  across  the  disc  from  east  to  west.  These  are 
evidently  attached  to  its  surface,  and  not  bodies  circling  around  the 
sun  at  a  distance  above  it,  as  was  imagined  by  some  early  astrono- 
mers, because,  as  Galileo  early  demonstrated,  they  continue  in  sight 
just  as  long  as  the  time  during  which  they  are  invisible  ;  which 
would  not  be  the  case  if  they  were  at  any  considerable  elevation. 

1  The  determination  of  the  sun's  density  does  not  necessarily  involve  its  parallax. 
Put  p  for  sun's  radius,  and  Ds  for  its  density  ;  let  De  be  earth's  mean  density. 


Substitute  in  equation  (c),  and  we  have  ±irp*Ds  =  ^irr^De  |~"^(~)(~)  1  >  whence 
Ds  —  ^j_"yi(")(~)  ]•  But  (;|)  =  sin  s>  s  teing  the  sun's  angular  semi- 
diameter.  Hence,  finally,  Da  =  .De 


PERIOD    OF   ROTATION. 


189 


Period  of  Rotation.  —  The  average  time  occupied  by  a  spot  in 
passing  around  the  sun  and  returning  to  the  same  position  again  is 
27.25  days,  —  average  because  different  spots  show  considerable 
differences  in  this  respect.  This  interval,  however,  is  not  the  true 
time  of  solar  rotation,  but  the  synodic,  since  the  earth  advances  in 
the  interval  of  a  revolution  so  that  the  sun  has  to  turn  on  its  axis  a 
little  farther  each  time  to  bring  the  spot  again  into  conjunction  with 
the  earth.  The  equation  by  which  the  true  period  is  deduced  from 
the  synodic  is  the  same  as  in  the  case  of  the  moon  (Art.  232),  viz.: 


T      E      S 

T  being  the  true  period  of  the  sun's  rotation,  E  the  length  of  the 
year,  and  S  the  observed  synodic  rotation  ; 

When°e>  = 


which  gives  jT=25d.35.     Different  observers  get  slightly  different 
results.     Carrington  finds  25d.38  ;  Spoerer,  25d.23. 

282.  Position  of  the  Sun's  Axis.  —  On  watching  the  spots  with 
care  as  they  cross  the  disc,  it  appears  that  they  usually  describe 
paths  more  or  less  oval,  showing  that  the  sun's  axis  is  inclined  to 
the  ecliptic.  Twice  a  year,  however,  the  paths  become  straight,  at 
the  times  when  the  earth  is  in  the  plane  of  the  sun's  rotation. 
These  dates  are  about  June  3  and  December  5. 


N 


N 


N 


March  1 


June  3  Sept.  4 

FIG.  90.  —  Position  of  the  Sun's  Axis. 


Dec.  5 


The  ascending  node  of  the  sun's  equator  is  in  celestial  longitude  73°  40' 
(Carrington),  and  the  inclination  of  its  equator  to  the  plane  of  the  ecliptic 
is  7°  15'.  Its  inclination  to  the  plane  of  the  terrestrial  equator  is  26°  25'. 
The  position  of  the  point  in  the  sky  towards  which  the  sun's  pole  is  directed 
is  in  right  ascension  18h  44m,  declination  +  63°  35',  very  nearly  half-way 
between  the  bright  star  a  Lyrse  and  the  Pole  Star. 

Fig.  90  shows  the  position  of  the  sun's  axis  and  equator  with  reference 
to  the  north  and  south  line,  and  the  apparent  paths  of  sun-spots  upon  the 


190  THE   SUN. 

disc,  at  the  dates  indicated.  On  January  4  and  July  6  the  axis  lies  exactly 
upon  the  hour-circle,  i.e.,  due  north  and  south  in  the  sky.  On  April  5  and 
October  14  the  position  angle  is  at  its  maximum  of  26°  25'  west  and  east, 
respectively. 

283.  Peculiar  Law  of  the  Sun's  Rotation.  —  Equatorial  Accelera- 
tion.    The  earth,  rotates  as  a  whole,  every  point  on  its  surface  making 
its  diurnal  revolution  in  the  same  time ;    so  also  with  the  moon  and 
with  the  planet  Mars.     Of  course  it  is  necessarily  so  with  any  solid 
globe.     But  this  is  not  the  case  with  the  sun.     It  was  noticed  quite 
early  that  the  different  spots  give  different  results  for  the  rotation 
period,  but  the  researches  of  Carrington  between  1853  and  1861  first 
brought  out  the  fact  that  the  differences  follow  a  regular  law,  showing 
that  at  the  solar  equator  the  time  of  rotation  is  less  than  on  either 
side  of  it.     Thus,  spots  near  the  sun's  equator  give  T=  25  days  ;  at 
solar  latitude  20°,  T=25.75  days;    at  solar  latitude  30°,  T=26.5 
days;  at  solar  latitude  40°,  T=27  days.     The  time  of  rotation  in 
latitude  40°  is  fully  two  days  longer  than  at  the  solar  equator  ;  but 
we  are  unable  to  follow  the  law  further  towards  the  poles,  because 
the  spots  are  rarely  found  beyond  the  parallels  of  45°  on  each  side 
of  the  equator,  and  there  are  no  well-defined  markings  between  this 
point  and  the  poles  by  which  we  can  accurately  determine  the  motion. 

284.  Various  formulse  have  been  proposed  to  represent  this  law  of  rota- 
tion.  Carrington  gives  for  the  daily  motion  of  a  spot  X=  865'  —  165'  X  sin*  /, 
/  being  the  solar  latitude  of  the  spot.     Faye,  from  the  same  observations, 
considering  that  the  exponent  \  could  have  no  physical  justification,  deduced 
X  =  862'  —  186'  X  sin2/,  which  agrees  almost  as  well  with  the  observations. 
Still  other  formulae  have  been  deduced  by  Spoerer,  Zollner,  and  Tisserand, 
all  giving  substantially  the  same  results  for  the  sun-spot  zones. 

It  might  be  supposed  that  this  apparent  equatorial  acceleration  may  be 
only  a  motion  of  the  spots  over  the  sun's  surface,  like  that  of  clouds  or  rail- 
way trains  over  the  earth,  and  the  idea  has  been  tested  by  observations 
upon  the  faculse  (Art.  292),  and  upon  the  lower  portions  of  the  solar  atmos- 
phere where  the  dark  lines  of  the  spectrum  originate.  The  results  from  the 
faculse  have  been  a  little  discordant  among  themselves,  but  a  late  research 
of  the  kind,  based  upon  a  series  of  photographs  made  at  Pulkowa,  comes 
out  in  substantial  agreement  with  the  results  obtained  from  the  spots. 

The  motion  of  the  sun's  atmosphere  cannot,  of  course,  be  studied  by 
direct  telescopic  or  photographic  methods,  but  only  spectroscopically,  as  ex- 
plained hereafter,  by  making  use  of  the  "  Doppler-Fizeau  Principle"  (Art. 
321).  The  earlier  observations  of  this  kind  were  not  delicate  enough  to  do 
much  more  than  to  prove  that  the  solar  atmosphere  actually  participates  in 
the  general  rotation.  In  1887  Crew  at  Baltimore  made  an  elaborate  series 
of  observations  which  indicated  for  the  atmosphere  a  mean  rotation-period 


THE  PHENOMENA  OF  THE  SUN'S  SURFACE.      191 

practically  the  same  as  that  given  by  the  spots,  but  with  a  slight  (though 
very  doubtful)  retardation  at  the  equator.  The  exquisite  work  of  Dune"r  (in 
Sweden),  however,  two  years  later,  demonstrated  the  equatorial  acceleration 
of  the  solar  atmosphere  beyond  all  question. 

Observations  of  Jewell  at  Baltimore  in  1897  appear  to  indicate  that  the 
upper  portions  of  the  solar  atmosphere  have  a  rotation-period  several  days 
shorter  than  the  lower.  Results  obtained  by  Adams  in  1907  also  indicate 
a  retardation  at  the  lower  level,  but  less  in  amount. 

285.  Thus  far  all  the  formulae  which  attempt  to  represent  the 
velocity  of  the  sun's  surface  in  different  latitudes  are  simply  empiri- 
cal;  that  is,  they  are  deduced  from,  the  observations,  without  being 
based  upon  any  satisfactory  physical  explanation,  for  no-  such  ex- 
planation of  this  strange  equatorial  acceleration  has  yet  been  found. 
Probably  it  has  its  origin  somehow  in  the  effects  produced  by  the 
outpour  of  heat  from  the  sun's  surface ;  still,  just  how  such  a  result 
should  follow  in  the  case  of  a  cooling  globe,  of  which  the  particles 
are  free  to  move  among  each  other,  is  not  yet  evident. 

(See  note  at  end  of  chapter,  Art.  310*.) 

It  has  been  suggested  (see  Art.  306)  that  the  spots  may  be  due  to  the  fall 
of  matter  upon  the  sun's  surface,  matter  which  has  remained  at  a  great 
elevation  for  some  time,  and  acquired  a  corresponding  velocity  of  rotation. 
It  can  be  shown  that  if  the  matter  forming  the  spots  had  thus  fallen  from 
a  height  of  about  20000  miles,  it  would  account  for  their  apparent  accelera- 
tion. Matter  so  falling  would  have  an  apparent  eastward  motion,  just  as 
do  bodies  on  the  earth  when  falling  from  the  summit  of  a  tower  (Art.  138). 
From  this  point  of  view  it  is  very  interesting  to  inquire  whether  the  minuter 
markings  upon  the  sun's  surface  do,  or  do  not,  possess  the  same  rate  of 
motion  as  the  spots.  At  present  the  evidence  is  not  decisive,  but  probably 
they  do. 

286.  The  Phenomena  of  the  Sun's  Surface.  —  In  order  to  study 
the    sun    with    the    telescope    it    is 

necessary  to  be  provided  with  some 
special  forms  of  apparatus.  Its  heat 
and  light  are  so  intense  that  it  is  im- 
possible to  look  directly  at  it,  as  we 
do  at  the  moon.  A  very  convenient 
method  of  exhibiting  the  sun  to  a 
number  of  persons  at  once  is  simply 
to  attach  to  the  telescope  a  frame 
carrying  a  screen  of  white  paper  at  a  FIG.  91.— Telescope  and  Screen, 

distance  of  a  foot  or  more  from  the 
eye-piece,  as  shown  in  Fig.  91.  On  pointing  the  instrument  to  the  sun 


192 


THE    SUN. 


and  properly  adjusting  the  focus,  a  distinct  image  is  formed  on  the 

screen,  which  shows  the  main  features 
very  fairly.  It  is,  however,  much  more 
satisfactory  to  look  at  it  directly,  with 
a  proper  eye-piece.  With  a  small 
telescope,  not  more  than  two  and  a  half 
or  three  inches  in  diameter,  a  mere 
dark  glass  between  the  eye-piece  and 
the  eye  can  be  used,  but  this  dark  glass 
soon  becomes  very  hot,  and  is  apt  to 
crack.  With  larger  instruments,  it  is 
necessary  to  use  eye-pieces  especially 
designed  for  the  purpose  and  known  as 
FIG.  92. —Herschei  Eye-piece.  solar  eye-pieces  or  helioscopes. 

The  simplest  of  them,  and  a  very  good  one  for  ordinary  purposes,  is  one 
known  as  Herschel's,  in  which  the  sun's  rays  are  reflected  at  right  angles  by  a 
plane  of  unsilvered  glass  (Fig.  92).  This  reflector  is  made  either  of  a  prismatic 
form  or  concave,  in  order  that  the  reflection  from  the  back  surface  may  not 
interfere  with  that  from  the  front.  About  nine-tenths  of  the  light  passes 
through  this  reflector,  and  is  allowed  to  pass  out  uselessly  through  the  open 
end  of  the  tube*.  The  remaining  tenth  is  sent  through  the  eye-piece,  and 
though  still  too  intense  for  the  eye 
to  endure,  it  requires  only  a  com- 
paratively thin  shade  of  neutral-tinted 
glass  to  reduce  it  sufficiently,  and  in 
this  case  the  shade  does  not  become 
uncomfortably  heated.  It  is  well  to 
have  the  shade-glass  made  wedge- 
shaped,  —  thinner  at  one  end  than 
at  the  other,  —  so  that  one  can 
choose  the  particular  thickness  which 
is  best  adapted  to  the  magnifying 
power  employed. 


287.  The  polarizing  eye-pieces 
are  still  better  when  well  made.  In 
these  the  light  is  reflected  twice  at 
plane  surfaces  of  glass  at  the  "  angle 
of  polarization"  (Physics,  p.  462), 
and  is  then  received  on  a  second  pair 
of  reflectors  of  black  glass.  When 
the  upper  pair  of  reflectors  is  in 
either  of  the  two  positions  shown  in 


FIG.  93.  —  Polarizing  Helioscope. 
Fig.  93,  a  strong  beam  of  light  is  received  at  C,  —  too  strong  for  the  eye  to 


PHOTOGRAPHY.  193 

bear,  although  more  than  ninety  per  cent  of  it  has  already  been  rejected  ; 
but  by  simply  turning  the  box  which  carries  the  upper  reflectors  one-quarter 
of  a  revolution  around  the  line  BB'  as  an  axis,  the  light  may  be  wholly 
extinguished ;  and  any  desired  gradation  may  be  obtained  by  setting  it  at 
the  proper  angle,  without  the  use  of  a  shade-glass. 

288.  It  may  be  asked  why  it  will  not  answer  merely  to  "  cap  " 
the  object-glass,  and  so  cut  off  part  of  the  light,  instead  of  rejecting 
it  after  it  has  once  been  allowed  to  enter  the  telescope.     It  is  because 
of  the  fact,  mentioned  in  Art.  43,  that  the  smaller  the  object-lens  of 
the  telescope,  the  larger  the  image  it  makes  of  a  luminous  point,  or 
the  wider  its   image  of  a  sharp  line.     To  cut  down  the  aperture, 
therefore,  is  to  sacrifice  the  definition  of  delicate  details.     With  a 
low  power  there  is  no  objection  to  reducing  the  amount  of  heat  ad- 
mitted into  the  telescope  tube  in  that  way,  but  with  the  higher 
powers  the  whole  aperture  should  always  be  used. 

289.  Photography.  —  In  the  study  of  the  sun's  surface  photog- 
raphy is  for  some  purposes  very  advantageous  and  much  used.    The 
instrument  must  have  a  special  object-glass  (Art.  42),  with  an  appa- 
ratus for  the  quick  exposure  of  plates.     Such  instruments  are  called 
photo-heliographs,  and  with  them  photographs  of  the  sun  are  made 
daily  at  numerous  observatories.      The  necessary  exposure  varies 
from  ^i^  to  -jL  of  a  second,  in  different  cases.     The  pictures  made 
by  these  instruments  are  usually  from  two  inches  up  to  eight  or  ten 
inches  in  diameter,  and  some  of  Janssen's,  made  at  Meudon,  bear 
enlarging  up  to  forty  inches  in  diameter.     Photographs  have  the 
advantage  of  freedom  from  prejudice  and  prepossession  on  the  part 
of  the  observer ;  but  they  take  no  advantage  of  the  instants  of  fine 
seeing.     They  represent  the  surface  as  it  happened  to  appear  at  the 
moment  when  the  plate  was  uncovered. 

290.  The  study  of  the  sun  has  become  so  important  from  a  scientific 
point  of  view  that  several  observatories  have   recently  been  established 
mainly  for  that  purpose,  though  most  of  them  connect  with  it  that  of  other 
topics  in  astronomical  physics.     Among  the  most  important  of  these  "  astro- 
physical"  observatories  may  be  named  those  at  Potsdam  and  Meudon,  and  in 
this  country  the  one  at  Mount  Wilson,  California. 

291.  General  Views.  —  Before  passing  to  a  discussion  of  the  de- 
tails of  the  different  solar  phenomena,  it  will  be  well  to  give  a  very 
brief  summary  of  the  objects  and  topics  to  be  considered. 

1.  The  photosphere  ;  i.e.,  the  luminous  surface  of  the  sun  directly 
visible  to  our  telescopes.  It  is  probably  a  sheet  of  luminous  clouds 
formed  by  condensation  into  little  drops  and  crystals  (like  the  water- 


194 


THE    SUN. 


drops  and  ice-crystals  in  our  terrestrial  clouds)  of  certain  substances 
which  within  the  central  mass  of  the  sun  exist  in  a  gaseous  form, 
but  are  cooled  at  its  surface  below  the  temperature  necessary  for 
their  condensation  ;  perhaps  such  substances  as  carbon,  boron,  and 
silicon.  The  granules,  faculse,  and  spots  are  all  phenomena  in  this 
photosphere. 

2.  The  so-called  "reversing  layer"  is  a  stratum  of  unknown  thick- 
ness, but  probably  shallow,  just  above  the  photosphere,  containing 
the  vapors  of  many  of  the  familiar  terrestrial  elements  ;  of  which 
the  presence,  and  to  some  extent  their  physical  condition,  can  be 
investigated  by  means  of  the  spectroscope. 

3.  Above  the  photosphere,  interpenetrating  the  atmosphere  of 
vapors  just  spoken  of,  and  perhaps  indistinguishable  from  it,  is  an 

envelope  of  permanent 
gases  ;  that  is,  gases 
which,  under  the  solar 
conditions,  cannot  be  con- 
densed into  clouds  of 
solid  or  liquid  particles. 
Among  them  hydrogen  is 
most  conspicuous.  This 
envelope  is  the  so-called 
Chromosphere;  and  from 
it  the  prominences  of 
various  kinds  rise,  some- 
times to  the  height  of 
hundreds  of  thousands 
of  miles.  These  beautiful 
objects  are  best  seen  at 
total  eclipses  of  the  sun, 
but  to  a  certain  extent 
they  can  also  be  studied 
at  any  time  by  the  help 
of  a  spectroscope. 

4.  Higher  yet  rises  the 
mysterious  Corona,  of 
material  still  less  dense, 
and  so  far  observable 
only  during  total  eclipses 
of  the  sun. 

Fig.  94  shows  the  relative  positions  of  these  different  elements  of 
the  solar  constitution. 


FIG.  94.  —  Constitution  of  the  Sun.    From  "  The  Sun,' 
by  permission  of  the  Publishers. 


THE   PHOTOSPHERE. 


195 


5.  A    fifth    subject    deals    with,    the   measurement  of  the   sun's 
light  and  the  relative   brightness   of  different  parts  of  the  solar 
surface. 

6.  Another  most  interesting  and  important  topic  relates  to  the 
amount  of  heat  radiated  by  the  sun,  —  the  sun's  probable  temperature 
and  the  mechanism  by  which  its  heat-supply  is  maintained. 

292.    The  Photosphere.  —  The  sun's  visible  surface  is  called  the 
photosphere,  and  when  studied  under  favorable  atmospheric  condi- 


FIG.  95. 

The  Great  Sun  Spot  of  September,  1870,  and  the  Structure  of  the  Photosphere.  From  a  Drawing 
by  Professor  Langley.  From  "  The  New  Astronomy,"  by  permission  of  the  Publishers. 

tions,  with  a  rather  low  magnifying  power,  it  looks  like  rough  draw- 
ing-paper. With  higher  powers  it  is  seen  to  be,  as  shown  in  Fig.  95, 
made  up  of  a  comparatively  darkish  background  sprinkled  over  with 
grains,  or  "  nodules,"  as  Herschel  called  them,  of  something  much 
more  brilliant,  —  like  snowflakes  on  gray  cloth,  according  to  Langley. 
These  are  from  400  to  600  miles  across,  and  in  the  finest  seeing  are 
themselves  resolved  into  more  minute  "granules."  Eor  the  most 


196  THE   SUN. 

part,  these  nodules  are  about  as  broad  as  they  are  long,  though  of 
irregular  form  ;  but  here  and  there,  especially  in  the  neighborhood 
of  the  spots,  they  are  drawn  out  into  long  streaks.  Nasmyth  seems 
first  to  have  observed  this  structure,  and  called  the  filaments  "  wil- 
low leaves."  Secchi  called  them  "  rice  grains."  According  to 
Huggins  they  were  "  dots  "  ;  and  there  was  for  a  long  time  a  pretty 
lively  controversy  as  to  their  true  form.  Their  shape,  however, 
unquestionably  varies  very  much  in  different  parts  of  the  surface 
and  under  different  circumstances.  They  are  probably  luminous 
clouds1  floating  in  a  less  luminous  atmosphere. 


FIG.  96.  —  Faculse  at  Edge  of  the  Sun.     (De  La  Hue.) 

Near  the  edge  the  photosphere  appears  generally  much  less  bril- 
liant ;  but  certain  bright  streaks  called  "  faculse"  (from  fax,  a  torch), 
which,  though  visible,  are  not  very  obvious  at  points  further  from 
the  limb,  become  there  conspicuous.  These  faculae  are  elevations, 
probably  of  the  same  material  as  the  rest  of  the  photosphere,  but 
elevated  above  the  general  level  and  intensified  in  brightness. 
When  one  of  them  passes  off  the  edge  of  the  sun,  it  is  sometimes 
seen  as  a  little  projection.  They  are  most  abundant  near  the  sun 
spots,  and  they  are  more  conspicuous  near  the  edge  of  the  disc,  as 
shown  in  Fig.  96,  because  the  sun's  surface  is  overlaid  by  a  gaseous 
atmosphere  which  absorbs  more  of  the  light  there  than  it  does  near 
the  centre,  and  these  faculae  push  up  through  it  like  mountains. 

1  It  was  many  years  ago  suggested  by  Stoney  that  these  clouds  are  probably  com- 
posed mainly  of  carbon,  but  this  view  is  not  yet  by  any  means  universally  accepted. 


THE   SUN   SPOTS.  197 

293.  The  Sun  Spots.  —  The  appearance  of  a  normal  sun  spot,  Fig. 
97,  fully  formed,  is  that  of  a  dark  central  " umbra"  more  or  less 
nearly  circular,  with  a  fringing  "penumbra"  composed  of  filaments 
directed  radially.  The  umbra  itself  is  not  uniformly  dark  through- 
out, but  is  overlaid  with  filmy  clouds  which  require  a  good  telescope 
and  helioscope  to  make  them  visible.  Usually,  also,  in  the  umbra 
there  are  several  round  and  very  black  spots,  which  are  sometimes 
called  "nucleoli,"  but  are  often  referred  to  as  "Dawes7  holes," 
after  the  name  of  their  first  discoverer.  But  while  this  is  the 
appearance  of  what  may  be  taken  as  a  normal  spot,  very  few  are 
strictly  normal.  Most  of  them  are  more  or  less  irregular  in  form. 
They  are  often  gathered  in  groups  with  a  common  penumbra,  and 
partly  covered  by  brilliant  "bridges"  extending  across  from  the 
outside  photosphere.  Often  the  umbra  is  out  of  the  centre  of  the 
penumbra,  or  has  a  penumbra  only  on  one  side,  and  the  penumbral 


FIG.  97.  —  A  Normal  Sun  Spot.    (SeccM;  modified.) 

filaments,  instead  of  being  strictly  radial,  are  frequently  distorted 
in  every  conceivable  way.  In  fact,  the  normal  spots  form  a  very 
small  proportion  of  the  whole  number. 

The  darkest  portions  of  the  umbra  are  dark  only  by  contrast. 
Photometric  observations  (by  Langley)  show  that  even  the  nucleus 
gives  at  least  one  per  cent  as  much  light  as  a  corresponding  area  of 
the  photosphere ;  that  is  to  say,  as  we  shall  see  hereafter,  the  dark- 
est portion  of  a  sun  spot  is  brighter  than  a  calcium  light. 


198 


THE   SUN. 


294.  The  spots  are  generally  believed  to  be  depressions  in  the 
photosphere,  filled  with  gases  and  vapors  which  are  cooler  than  the 
surrounding  portions,  and  therefore  absorb  a  considerable  proportion 
of  light.  The  evidence  that  they  are  "hollows"  is  the  change 
in  the  appearance  of  a  spot  as  it  travels  across  the  disc.  According 
to  Wilson  of  Glasgow,  who  first  discovered  the  fact  (if  it  really  is 
one)  more  than  a  century  ago,  the  umbra  of  a  normal  spot  is  central 
at  the  centre  of  the  disc,  but  as  the  spot  approaches  the  limb  the 
penumbra  becomes  narrower  on  the  inner  edge,  and  vanishes  entirely 
before  the  spot  disappears  around  the  limb,  —  the  appearance  (Fig. 
98)  being  precisely  such  as  would  be  shown  by  a  saucer-shaped  cav- 
ity in  the  surface  of  a  globe  if  the  bottom  of  the  cavity  were  painted 
black  to  represent  the  umbra,  and  the  sloping  sides  gray  for  the 
penumbra.  Evidently  observations  upon  any  single  spot  would  be 
inconclusive,  because  spots  are  extremely  irregular  in  form  and  be- 
havior ;  but  by  observing  several  hundred  the  truth  ought  to  come 


FIG.  98.  —  Sun  Spots  as  Cavities. 

out  distinctly,  and  until  recently  astronomers  were  practically  unani- 
mous in  accepting  Wilson's  theory.  Lately,  however,  some  high 
authorities  have  called  it  in  question ;  partly  on  the  evidence  drawn 
from  solar  photographs  and  a  very  extensive  series  of  sun-spot 
drawings  by  Mr.  Howlett,  an  observer  of  great  experience,  and 
partly  on  account  of  certain  thermal  observations  referred  to  in 
Art.  301*.  The  subject  is  still  under  discussion,  but  it  seems  now 
most  probable  that  different  spots  lie  at  very  different  levels,  some 
low  down,  depressions  in  the  photosphere,  but  others  at  a  consid- 
erable elevation  above  it. 

295.  The  penumbra  is  usually  composed  of  "  thatch-straws,"  or 
long  drawn-out  granules  of  photospheric  matter,  which,  as  has  been 
said,  converge  in  a  general  way  towards  the  centre  of  the  spot.  At 
the  inner  edge  the  penumbra,  from  the  convergence  of  these  filaments, 
is  usually  brighter  than  the  outer.  The  inner  ends  of  the  filaments 
are  generally  club-formed ;  but  sometimes  they  are  drawn  out  into 


DIMENSIONS  OF  SUN  SPOTS.  199 

fine  points,  which  seem  to  curve  downward  into  the  umbra  like 
the  rushes  over  a  pool  of  water.  The  outer  edge  of  the  penum- 
bra is  usually  pretty  definite,  and  the  penumbra  there  is  darker. 
Around  the  spot  the  photosphere  is  much  disturbed  and  elevated 
into  faculse,  which  sometimes  radiate  outward  from  the  spot  like 
streams  of  lava  from  a  crater,  though,  of  course,  they  are  really 
nothing  of  the  sort. 

296.  Dimensions  of  Sun  Spots.  —  The  diameter  of  the  umbra  of 
a  sun  spot  ranges  all  the  way  from  500  to  1000  miles  in  the  case  of 
a  very  small  one,  to  50000  or  60000  miles  in  the  case  of  the  larger 
ones.     The  penumbra  surrounding  a  group  of  spots  is  sometimes 
150000  miles  across,  though  that  would  be  rather  an  exceptional 
size.     Not  infrequently  sun  spots  are  large  enough  to  be  seen  by 
the  naked  eye  through  a  fog  or  with  the  help  of  a  colored  glass. 

The  Chinese  have  many  records  of  such  objects,  but  the  real  dis- 
covery of  sun  spots  dates  from  1610,  as  an  immediate  consequence 
of  Galileo's  discovery  of  the  telescope. 

297.  Development  and  Changes  of  Form.  —  Generally  the  origin  of 
a  sun  spot  fails  to  be  observed.     It  begins  from  an  insensible  point, 
and  rapidly  grows  larger,  the  penumbra  usually  appearing  only  after 
the  nucleus  is  fairly  developed. 

If  the  disturbance  which  causes  the  spot  is  violent,  the  spot  usually 
breaks  up  into  several  fragments,  and  these  again  into  others  which 
tend  to  separate  from  each  other.  At  each  new  disturbance  the  for- 
ward portions  of  the  group  show  a  tendency  to  advance  eastward  on 
the  sun's  surface,  leaving  behind  them  a  trail  of  smaller  spots. 

298.  The  "segmentation"  of  a  spot,  as  Faye  calls  it,  is  usually  effected 
by  the  formation  of  a  "bridge,"  or  streak  of  brilliant  light,  which  projects 
itself  across  the  penumbra  and  umbra  from  the  outside  photosphere.     These 
bridges  are  mere  extensions  of  the  surrounding  faculse,  and  are  often  intensely 
bright. 

Occasionally  a  spot  shows  a  distinct  cyclonic  motion,  the  filaments  being 
drawn  inward  spirally  ;  and  in  different  members  of  the  same  group  of  spots 
the  cyclonic  motions  are  not  seldom  in  opposite  directions. 

When  a  spot  at  last  vanishes  it  is  usually  by  the  rapid  encroachment  of 
the  photospheric  matter,  which,  as  Secchi  expresses  it,  appears  to  "fall  pell- 
mell  into  the  cavity,"  completely  burying  it  and  leaving  its  place  covered 
by  a  group  of  f  aculae.  Figs.  99-104  (see  page  201)  show  the  changes  which 
took  place  in  the  great  spot  of  September,  1870.  They  are  from  photographs 
by  Mr.  Rutherfurd  of  New  York,  and  are  borrowed  from  "  The  New  Astron- 
omy "  of  Professor  Langley,  through  the  courtesy  of  his  publishers. 


200 


THE  SUN. 


Spots  within  15°  or  20°  of  the  sun's  equator  generally,  on 
the  whole,  drift  a  little  towards  it,  while  those  in  higher  latitudes 
drift  away  from  it ;  but  the  motion  is  slight,  and  exceptions  are  fre- 
quent. 

In  and  around  the  spot  itself  the  motion  is  usually  inward  towards 
the  centre,  and  downward  at  the  centre.  Not  infrequently  the  frag- 
ments at  the  inner  end  of  the  penumbral  filaments  appear  to  draw 
off,  move  towards  the  centre  of  the  spot,  and  then  descend.  Occa- 
sionally, though  seldom,  the  motion  is  vigorous  enough  to  be  detected 
by  the  displacement  of  lines  in  the  spectrum. 

300.  Duration. — The  duration  of  the  spots  is  very  various,  but, 
astronomically   speaking,  they   are   always   short-lived    phenomena, 
sometimes  lasting  for  only  a  few  days,  more  frequently,  perhaps,  for 
a  month  or  two.     In  a  single  instance,  a  spot  has  been  observed 
through  as  many  as  eighteen  successive  revolutions  of  the  sun. 

301.  Distribution.  — It  is  a  significant  fact  that  the  spots  are  con- 
fined mostly  to  two  zones  of  the  sun's  surface  between  5°  and  40°  of 
latitude  north  and  south.     A  few  are  found  near  the  equator,  none 


FIG.  105.  — Distribution  of  Sun  Spots  in  Latitude. 

beyond  the  latitude  of  45°.     Fig.  105  shows  the  distribution  of  sev- 
eral thousand  spots  as  observed  by  Carrington  and  Sporer. 

Occasionally,  what  Trouvelot calls  "veiled  spots"  are  seen  beyond 
the  45°  limits  —  grayish  patches  surrounded  by  faculae,  which  look  as 
if  a  dark  mass  were  submerged  below  the  surface  and  dimly  seen 
through  a  semi-transparent  medium. 


SUN   SPOTS. 


201 


FIG.  101.  —  Sept.  21. 


FIG.  100.  —  Sept.  20. 


FIG.  102.— Sept.  22. 


FIG.  103.— Sept  23.  FIG.  104.  —  Sept.  26. 

The  Great  Sun  Spot  of  1870. 


202  THE   SUN. 

301*.  Radiation  and  Temperature  of  Sun  Spots.  —  Thermopile  ob- 
servations upon  sun  spots,  first  made  in  1845  by  Henry,  and  since 
then  by  numerous  other  observers,  show  that  as  the  spots  are  darker, 
so  also  they  radiate  less  heat  than  other  portions  of  the  solar  sur- 
face. But  while  the  umbra  of  a  spot  generally  emits  less  than  one 
per  cent  as  much  light,  it  ordinarily  radiates  nearly  fifty  per  cent  as 
much  heat  as  an  equal  area  of  the  neighboring  photosphere,  and  if 
the  spot  is  near  the  edge  of  the  disc  the  percentage  rises  much  higher ; 
indeed,  Langley  and  Frost  have  met  with  cases  where  the  radiation 
of  a  spot  has  appeared  actually  to  exceed  that  of  the  brighter  regions 
surrounding  it,  —  an  important  and  rather  perplexing  observation. 

It  has  generally  been  inferred  hitherto  that  the  lower  heat  radia- 
tion of  a  sun  spot  indicates  a  lower  temperature  than  that  of  the 
surrounding  photosphere,  but  this  does  not  necessarily  follow.  Two 
masses  in  contact  and  at  the  same  temperature,  but  of  different  con- 
stitution, may  differ  widely  both  in  luminosity  and  in  their  radiating 
power  for  the  invisible  rays ;  for  instance,  the  mantle  and  the  gas- 
flame  of  a  Welsbach  burner.  At  present  it  is  perhaps  still  uncertain 
whether  the  spots  are  cooler  or  warmer  than  the  photospheric 
«  mantle.7' 

302.  Theories  as  to  the  Nature  of  the  Spots.  —  We  first  mention 
(a)  the  theory  of  Sir  William  Herschel,  because  it  still  finds  place 
in  certain  text-books,  though  certainly  incorrect.     His  belief  was 
that  the  spots  were  openings  through  two  luminous  strata,  which  he 
supposed  to  surround  the  central  globe  of  the  sun.     This  globe  he 
supposed  to  be  dark  (and  even  habitable  /).     The  outer  stratum,  the 
photosphere,  was  the  brighter  of  the  two,  and  the  opening  in  it  the 
larger,  while  the  inner  shell  between  it  and  the  solid  globe  was  of 
less  luminous  substance,  and  formed  the  penumbra.     He  thought  the 
opening  through  these  might  be  caused  by  volcanoes  on  the  globe 
beneath. 

303.  (b)  Another  theory,  now  generally  abandoned,  but  recently 
endorsed  by  Proctor  in  his  "  Old  and  New  Astronomy,"  was  pro- 
posed independently  both  by  Secchi  and  Faye  about  1868.     They 
supposed  that  the  spots  were  openings  in  the  photosphere  caused 
by  the  bursting  outward  of  the  imprisoned  gases  underneath  it. 

They  explained  the  darkness  of  the  centre  of  the  spot  by  the  fact  that  a 
heated  gas  at  a  given  temperature  has  a  lower  radiating  power  and  sends 
out  much  less  light  than  a  liquid  surface,  or  than  clouds  formed  by  the  con- 
densation of  the  same  material  at  even  a  lower  temperature.  This  is  true 


NATURE    OF   THE   SPOTS.  203 

of  gases  at  low  pressure,  but  not  of  gases  under  great  compression,  such  as 
must  be  the  case  within  the  body  of  the  sun.  Besides,  if  the  gases  possessed 
the  small  radiating  power  necessary  to  this  theory,  they  would  also  possess 
small  absorbing  power,  and  therefore  would  be  transparent ;  the  inner  side  of 
the  photosphere  on  the  opposite  side  of  the  sun  would  therefore  be  visible 
through  the  opening,  so  that  the  centre  of  such  an  eruption  would  not  be 
dark,  but,  if  anything,  brighter  than  the  general  solar  surface.  Moreover, 
as  we  now  know  from  the  spectroscopic  evidence,  the  motion  at  the  centre 
of  a  spot  is  usually  inward,  not  outward. 

304.  (c)  Faye  more  recently  has  proposed  and  now  maintains  a 
theory  which  has  numerous  good  points  about  it,  and  is  accepted  by 
many;  viz.,  that  the  spots  are  analogous  to  storms  on  the  earth,  being 
cyclones,  due  to  the  fact  that  the  portions  of  the  sun's  surface  near 
the  equator  make  their  revolution  in  a  shorter  time  than  those  in 
higher  latitudes.     This  causes  a  relative  drift  in  adjacent  portions 
of  the  photosphere,  and  according  to  him  gives  rise  to  vortices  or 
whirlpools  like  those  in  swiftly  running  water.     The  theory  explains 
the  distribution  of  the  spots  (which  abound  precisely  in  the  regions 
where  this  relative  drift  is  at  the  maximum)  and  many  other  facts, 
such  as  their  "  segmentation."     According  to  it,  however,  all  spots 
should  be  cyclonic,  and  the  spiral  motion  of  all  the  spots  in  the 
southern  hemisphere  should  be  clock-wise,  while  in  the  northern 
hemisphere  they  should  be  counter-clock-wise.     Now,  as  a  matter  of 
fact,  only  a  very  few  of  the  spots  show  such  spiral  motions,  and 
there  is  no  such  agreement  in  the  general  direction  of  the  motion 
as  the  theory  requires. 

Faye  attempts  to  account  for  this  by  saying  that  we  do  not  see  the  vortex 
itself,  but  only  the  cloud  of  cooler  materials  which  is  drawn  together  by  the 
down-rushing  vortex,  itself  hidden  beneath  this  cloud.  Still,  it  would  seem 
that  in  such  a  case  the  cloud  itself  should  gyrate.  Moreover,  the  relative 
drift  of  the  adjacent  portions  of  the  photosphere  is  too  small  to  account  for 
the  phenomena  satisfactorily.  In  the  solar  latitude  of  20°  two  points  sepa- 
rated by  1'  of  the  sun's  surface  (123  miles)  have  a  relative  daily  drift  of  only 
about  four  and  one-sixth  miles,  insufficient  to  produce  any  sensible  whirling. 

305.  (d)  Secchi's  later  theory.    He  supposed  the  spots  to  be  due 
to  eruptions  from  the  inner  portions  of  the  sun's  surface,  not  in  the 
spot,  however,  but  only  near  it  ;  the  spot  itself  being  formed  by  the 
settling  down  upon  the  photosphere  of  materials  thrown  out  by 
the  eruption  and  cooled  by  their  expansion  and  their  motion  through 
the  upper  regions.     We  have,  however,  in  fact,  as  a  usual  thing, 


204  THE   SUN. 

not  a  single  eruption,  but  a  ring  of  eruptions  all  around  every  large 
spot,  all  of  them  converging  their  bombardment,  so  to  speak,  upon 
the  same  centre,  —  a  fact  very  difficult  to  explain  if  the  spot  origi- 
nates in  the  eruption,  but  not  difficult  to  understand  if  the  eruptions 
are  the  result  of  the  spot. 

Possibly  the  true  explanation  may  be  that  when  an  eruption  occurs 
at  any  point,  the  photosphere  somewhere  in  the  neighborhood  settles 
down  in  consequence  of  the  diminution  of  the  pressure  beneath,  thus 
forming  a  "sink"  so  to  speak,  which  is  of  course  covered  by  a  greater 
depth  of  absorbing  vapors  above,  and  so  looks  dark. 

306.  (e)  Mr.  Lockyer,  in  his  recent  work  on  the  "Chemistry  of 
the  Sun,"  revives  an  old  theory,  first  suggested  by  Sir  John  Herschel 
and  accepted  by  the  late  Professor  Peirce,  that  the  spots  are  not 
formed  by  any  action  from  within,  but  by  cool  matter  descending  from 
above,  —  matter  very  likely  of  meteoric  origin ;  but  it  is  difficult  to 
see  how  the  distribution  of  the  spots  with  reference  to  the  sun's 
equator  can  be  accounted  for  in  this  way. 

Schaeberle,  of  the  Lick  Observatory,  accounts  for  them  in  a  somewhat 
similar  manner,  but  considers  that  the  descending  streams  consist  of  matter 
returning  to  the  sun  after  having  been  projected  from  it  by  some  repulsive 
force  to  distances  of  hundreds  of  millions  of  miles.  See  Art.  331. 

306*.  (/)  The  newest  theory  of  those  that  deserve  any  considera- 
tion is  one  proposed  by  E.  Oppolzer,  of  Vienna,  in  1893,  and  is  based 
largely  on  the  recent  researches  of  meteorologists  upon  the  thermal 
effects  of  vertical  currents  in  our  own  atmosphere.  Such  currents 
are  supposed  to  rise  periodically  from  the  polar  regions  of  the  sun, 
to  drift  slowly  toward  its  equator,  and  to  descend  in  the  spot-zones, 
becoming  heated  and  "dried"  in  their  descent,  thus  forming  in  the 
photosphere  hollows  which  are  filled  with  metallic  vapors  in  a  purely 
gaseous  condition.  According  to  this  theory,  the  temperature  of  a 
spot  is  higher  than  that  of  the  surrounding  medium.  In  many  ways 
the  theory  corresponds  admirably  with  facts,  explaining  better  than 
any  other  the  peculiar  character  of  the  sun-spot  spectrum  (Art.  321), 
Spoerer's  law  of  sun-spot  latitudes  (Art.  307*),  and  the  otherwise 
puzzling  observations  of  Langley  and  Frost  referred  to  in  Art. 
301*.  But  the  "periodical  polar  streams"  remain  themselves  to  be 
accounted  for. 

On  the  whole,  it  is  impossible  to  say  that  the  problem  of  the  origin 
of  sun  spots  is  yet  satisfactorily  solved.  There  is  no  question  that 


PERIODICITY    OF    SUN    SPOTS. 


205 


sun  spots  are  closely  associated  with  eruptions  from  beneath;  but 
which  is  cause  and  which  effect,  or  whether  both  are  due  to  some 
external  action,  remains  undetermined. 

307.  Periodicity  of  Sun  Spots.  —  In  1843  Schwabe,  of  Dessau,  by 
the  comparison  of  an  extensive  series  of  observations  running  over 
nearly  thirty  years,  showed  that  the  sun  spots  are  periodic,  being  at 
times  vastly  more  numerous  than  at  others,  with  a  roughly  regular 
recurrence  every  ten  or  eleven  years.  This  had  been  surmised 
by  Horrebow  more  than  a  century  before,  though  not  proved. 


ir,o 


100 


100 


FIG.  106.  —  Wolf's  Sun-spot  Numbers. 


Subsequent  study  fully  confirms  this  remarkable  result  of  Schwabe. 
Wolf  of  Zurich  has  collected  all  the  observations  discoverable  and 
finds  a  pretty  complete  record  back  to  1610.  From  these  records  is 
constructed  the  annexed  diagram,  Fig.  106.  The  ordinates  of  the 
curve  represent  what  Wolf  calls  his  "relative  numbers,"1  which  he 
has  adopted  as  representing  the  spottedness. 

1  This  "relative  number"  is  formed  in  rather  an  arbitrary  manner  from  the 
observations  which  Wolf  hunted  up  as  the  basis  of  his  investigation.  The 
formula  is,  r  (the  relative  value)  ==  k  (10  g  -f  /),  in  which  g  is  the  number 
of  groups  and  isolated  spots  observed,/  the  total  number  of  spots  which  can 
be  counted  in  these  groups  and  singly,  while  k  is  a  coefficient  which  depends 
upon  the  observer  and  the  size  of  his  telescope  ;  it  is  large  for  a  small  telescope 


206 


THE   SUN. 


The  average  period  is  eleven  and  one-tenth  years,  but,  as  the  figure 
shows,  the  spot  maxima  are  quite  irregular,  both  in  time  and  extent. 
The  last  two  maxima  occurred  in  1893  and  in  1905.  Both  were 
comparatively  feeble ;  84  and  64  respectively  on  scale  of  Fig.  106. 
During  a  maximum  the  surface  of  the  sun  is  never  free  from  spots, 
from  twenty-five  to  fifty  being  frequently  visible  at  once.  During 
a  minimum,  on  the  other  hand,  weeks  often  pass  without  the  appear- 
ance of  a  single  one. 

307*.  Spoerer's  Law  of  Sun-spot  Latitudes —  Still  another  fact,  as 
yet  unexplained,  and  probably  of  great  theoretical  importance,  has  recently 
been  brought  out  by  Spoerer.  Speaking  broadly,  the  disturbance  which 
produces  the  spots  of  a  given  sun-spot  period  first  manifests  itself  in  two 
belts  about  30°  north  and  south  of  the  sun's  equator.  These  belts  then 
draw  in  toward  the  equator,  and  the  sun-spot  maximum  occurs  when  their 
latitude  is  about  16°  ;  while  the  disturbance  gradually  and  finally  dies  out 
at  a  latitude  of  8°  or  10°,  some  twelve  or  fourteen  years  after  its  first  out- 
break. Two  or  three  years  before  this  disappearance,  however,  two  new 
zones  of  disturbance  show  themselves.  Thus,  at  the  sun-spot  minimum  there 
are  four  well-marked  spot-belts  ;  two  near  the  equator,  due  to  the  expiring 
disturbance,  and  two  in  high  latitudes,  due  to  the  newly  beginning  outbreak ; 


18 

55                 18 

60                18 

65                 18 

70                   18 

75                  18 

80 

L. 

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W. 

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0 

FIG.  106*.  —  Spoerer's  Curves  of  Sun-spot  Latitude. 

and  it  appears  that  the  true  sun-spot  cycle  is  from  twelve  to  fourteen  years 
long,  each  beginning  in  high  latitudes  before  the  preceding  one  has  expired 
near  the  equator. 

Fig.  106*  illustrates  this,  embodying  Spoerer's  results  from  1855  until 
1880.  The  dotted  curves  show  Wolf's  sun-spot  curve  for  that  period,  the 
vertical  column  at  the  right  of  the  figure,  marked  W  at  the  top,  giving 
Wolf's  "  relative  numbers.'"  The  two  continuous  curves,  on  the  other  hand, 
give  the  solar  latitudes  of  the  two  series  of  spots  that  invaded  the  sun's  sur- 


and  not  very  persistent  observer,  and  approaches  unity  in  proportion  to  the  prob- 
able ratio  between  the  actual  total  number  of  visible  spots  and  the  number  which 
the  observer  has  recorded. 


TERKESTRIAL   INFLUENCE   OF   SUN   SPOTS.  207 

face  in  those  years,  the  scale  of  latitudes  being  on  the  left  hand.  The  first 
series  began  in  1856  and  ended  in  1868  ;  the  second  broke  out  in  1866  and 
lasted  until  1880. 

308.  Possible  Cause  of  the  Periodicity The  cause  of  sun-spot 

periodicity  is  not  yet  known.     Attempts  have  been  made  to  account  for  it 
by  planetary  influences,  but  with  very  doubtful  success.     Sir  John  Herschel 
suggested  meteoric  swarms  moving  in  oval  orbits  with  an  eleven-year  period, 
and  with  a  perihelion  distance  so  small  that  many  of  the  meteors  strike  the 
sun's  surface  when  passing  perihelion  ;  an  idea  still  favored  by  Lockyer  and 
some  other  authorities.     Probably,  however,  the  most  general  impression  is 
that  this  rather  irregular  periodicity  is  more  likely  to  be  due  not  to  external 
causes  at  all,  but  to  something  in  the  constitution  of  the  photosphere  and 
the  rate  at  which  the  sun  is  losing  heat :  a  gathering  of  deep-lying  forces 
during  a  period  of  outward  quiescence,  followed  by  an  outburst  which  relieves 
the  internal  strain. 

309.  Terrestrial  Influence  of  the  Sun  Spots.  —  One  correlation 
of  sun  spots  with  the  earth  is  perfectly  demonstrated.     When  the 
spots  are  numerous,  magnetic  disturbances  (the  so-called  magnetic 
storms)  are  most  numerous  and  violent  upon  the  earth,  a  fact  not  to 
be  wondered  at  since  violent  disturbances  upon  the  sun's  surface 

^have  been  in  many  individual  cases  immediately  followed  by  mag- 
netic storms,  with  a  brilliant  exhibition  of  the  Aurora  Borealis. 
The  nature  and  mechanism  of  the  connection  is  as  yet  unknown,  but 
of  the  fact  there  can  be  no  question.  The  dotted  lines  in  the  figure 
of  the  sun-spot  periodicity  (Fig.  106)  represent  the  magnetic  stormi- 
ness  of  the  earth  at  the  indicated  dates ;  and  the  correspondence 
between  these  curves  and  the  curve  of  spottedness  makes  it  impos- 
sible to  doubt  the  connection. 

310.  It  has  been  attempted,  also,  to  show  that  greater  or  less 
disturbance  of  the  sun's  surface,  as  indicated  by  the  greater  fre- 
quency of  the  sun's  spots,  is  accompanied  by  effects  upon  the  mete- 
orology of  the   earth,  upon  its  temperature,   barometric   pressure, 
storminess,  and  the  amount  of  rainfall.     The  researches  of  Mr.  Mel- 
drum  of  Mauritius  with  respect  to  the  cyclones  in  the  Indian  Ocean 
appear  to  bear  out  the  conclusion  that  there  may  be  some  such  con- 
nection in  that  case,  but  the  general  results  are  by  no  means  decisive. 
In  some  parts  of  the  earth  the  rainfall  seems  to  be  greater  during  a 
spot  maximum ;  in  others,  less. 

As  to  the  temperature,  it  is  still  uncertain  whether  it  is  higher  or 
lower  at  the  time  of  a  spot  maximum.     The  spots  usually  give  less  heat 


208  THE   SUN. 

(as  Henry,  Secchi,  and  Langley  have  shown)  than  the  general  sur- 
face of  the  photosphere  ;  but  their  extent  is  never  sufficient  to  reduce 
the  amount  of  heat  radiated  from  the  sun  by  as  much  as  TTyV<r  Part- 
On  the  other  hand,  when  the  spots  are  most  numerous,  the  generally 
disturbed  condition  of  the  photosphere  would,  as  Langley  has  shown, 
necessarily  be  accompanied  by  an  increased  radiation. 

Dr.  Gould  considers  that  the  meteorological  records  in  the  Argen- 
tine Kepublic  between  1875  and  1885  show  an  indubitable  connection 
between  the  wind  currents  and  the  number  of  sun  spots.  But  the 
demonstration  of  such  a  relation  really  requires  observations  running 
through  several  spot  periods.  On  the  whole,  it  is  now  quite  certain 
that  whatever  influence  the  sun  spots  exert  upon  terrestrial  mete- 
orology is  very  slight,  if  it  exists  at  all. 

310*.    Possible  Explanation  of  the  Sun's  Equatorial  Acceleration. 

—  Recent  investigations  by  Wilsing  have  led  him  to  the  conclusion  that  the 
peculiar  and  perplexing  equatorial  acceleration  of  the  sun  depends  upon  its 
being  in  a  transitional  condition  between  a  nebula  and  a  solidified  globe. 
It  is  tending  towards  a  rotation  uniform  throughout,  and  will  ultimately 
reach  that  state  when  the  relative  motions  of  different  portions  of  its  mass 
have  been  destroyed,  as  they  will  be,  by  internal  friction.  Probably  they 
have  already  practically  disappeared  in  the  sun's  interior,  though  still  per- 
sisting at  and  near  its  surface.  They  die  out  so  slowly,  however,  that  it 
must  require  thousands,  if  not  millions,  of  years  to  make  them  vanish  en- 
tirely, and  through  any  short  period  of  a  century  or  two  they  appear  to  us  as 
permanent.  Their  explanation  lies,  therefore,  not  in  the  present  constitution 
of  the  sun,  but  in  its  past  history. 

This  view  is  substantially  confirmed  by  an  elaborate  mathematical  investi- 
gation by  Professor  Sampson  of  Durham.  University,  published,  in  the  last 
(51st)  volume  of  the  Memoirs  of  the  Royal  Astronomical  Society.  Both 
•writers  concur  in  the  conclusion  that  the  present  laws  of  surface  drift  upon 
the  sun,  as  shown  by  the  observations  of  Carrington,  Dune'r,  and  others, 
are  not  only  possible,  but  extremely  probable  (though  only  temporary)  con- 
sequences of  the  sun's  slow  condensation  from  a  nebulous  mass. 

Still  more  recently  (1901),  however,  Eniden  of  Munich  has  attempted  to 
sliow  that  the  acceleration  may  be  mathematically  explained  as  a  necessary 
consequence  of  physical  laws  applicable  to  a  gaseous  globe  rotating  and 
losing  heat  by  radiation  from  its  surface.  The  question  can  hardly  be  con- 
sidered as  finally  settled  even  yet. 


THE   SPECTROSCOPE.  209 


CHAPTER   IX. 

THE  SPECTROSCOPE  AND  THE  SOLAR  SPECTRUM.  —  CHEMICAL 
ELEMENTS  PRESENT  IN  THE  SUN. THE  SUN-SPOT  SPEC- 
TRUM. —  DOPPLER'S  PRINCIPLE.  —  THE  CHROMOSPHERE.  - 

THE   SOLAR   PROMINENCES.  —  THE   CORONA. 

311.  ABOUT  1860  the  spectroscope  appeared  in  the  field  as  a  new 
and  powerful  instrument  of  astronomical  research,  at  once  resolving 
many  problems  as  to  the  nature  and  constitution  of  the  heavenly 
bodies  which  before  had  not  seemed  to  be  even  open  to  investigation. 
It  is  hardly  extravagant  to  say  that  its  invention  has  done  for 
astronomy  almost  as  much  as  the  invention  of  the  telescope.  The 
latter  brings  distant  objects  optically  nearer  and  enables  us  to 
determine  their  positions  in  the  heavens  with  an  accuracy  before 
impossible,  and,  in  the  case  of  the  sun,  moon,  and  planets,  to  study 
their  forms  and  surface  markings.  It  reveals  also  millions  of  stars 
and  nebulae  otherwise  invisible. 

The  spectroscope,  on  the  other  hand,  enables  us  to  study  the  light 
itself,  and  to  read  in  it  a  record,  more  or  less  complete,  of  the 
chemical  constitution  and  physical  condition  of  the  body  from 
which  the  light  proceeds,  and  to  measure  the  rate  of  its  motion 
towards  or  from  us. 

The  essential  part  of  the  apparatus  is  either  a  prism  or  train  of 
prisms,  or  else  a  diffraction  grating,1  which  is  capable  of  performing 
the  same  office  of  dispersing  —  that  is,  of  spreading  and  sending  in 
different  directions  —  the  light  rays  of  different  wave-lengths.  If, 
with  such  a  "  dispersion  piece"  as  it  may  be  called  (either  prism  or 
grating),  one  looks  at  a  distant  point  of  light,  as  a  star,  he  will  see 
instead  of  a  point  a  long  streak  of  light,  red  at  one  end  and  violet 
at  the  other.  If  the  object  looked  at  be  not  a  point,  but  a  line  of 
light  parallel  to  the  edge  of  the  prism  or  to  the  lines  of  the  grating, 
then,  instead  of  a  mere  colored  streak  without  width,  one  gets  a 

1  The  grating  is  merely  a  piece  of  glass  or  speculum  metal,  ruled  with  many 
thousand  straight,  equidistant  lines,  from  5000  to  20000  in  the  inch.  Usually 
the  surface  before  ruling  is  accurately  plane,  but  for  some  purposes  the  concave 
gratings,  originated  by  Professor  Rowland,  are  preferable. 


210 


THE   SUN. 


spectrum,  a  colored  band  of  light,  which  may  show  markings  that 
will  give  the  observer  most  valuable  information.  (Physics,  pp.  444, 
450-451.)  For  convenience  sake  it  is  usual  to  form  this  line  of 
light  by  admitting  the  light  through  a  narrow  " slit"  which  is  at 
one  end  of  a  tube  having  at  the  other  end  an  achromatic  object-glass 
at  such  a  distance  that  the  slit  is  in  its  principal  focus.  This  tube 
with  slit  and  lens  constitutes  the  "  collimator"  so  called  because  it 
is  precisely  the  same  as  the  instrument  used  in  connection  with  the 
transit  instrument  to  adjust  its  line  of  collimation  (Art.  60). 

Instead  of  looking  at  the  spectrum  with  the  naked  eye,  however, 
it  is  better  in  most  cases  to  use  a  small  telescope ;  called  the  "view- 
telescope"  to  distinguish  it  from  the  large  telescope,  to  which  the 
spectroscope  is  often  attached. 


Prism-Spectroscope 


Orating 


Direct- Vision  Spectroscope 
FIG.  107.  —  Different  Forms  of  Spectroscope. 

312.  Construction  of  the  Spectroscope.  —  The  instrument,  there- 
fore, as  usually  constructed,  and  shown  in  Fig.  107,  consists  of  three 
parts,  —  collimator,  dispersion-piece,  and  view-telescope ;  but  in  the 
direct-vision  spectroscope,  shown  in  the  figure,  the  view-telescope  is 
omitted.  If  the  slit,  S,  be  illuminated  by  strictly  homogeneous 
light  all  of  one  wave-length,  say  yellow,  the  "real  image"  of  the 
slit  will  be  found  at  Y.  If  at  the  same  time  light  of  a  different 
wave-length  be  also  admitted,  say  red,  a  second  image  will  be  formed 
at  R,  and  the  observer  will  see  a  spectrum  with  two  "  bright  lines/' 


THE   SPECTROSCOPE.  211 

the  lines  being  really  nothing  more  than  images  of  the  slit.  If  light 
from  a  can<lle  be  admitted,  there  will  be  an  infinite  number  of  these 
slit-images,  close  packed,  like  the  pickets  in  a  fence,  without  inter- 
val or  break,  and  we  then  get  a  "  continuous "  spectrum.  If  we 
look  at  the  light  emitted  by  a  so-called  Geissler-tube,  containing 
rarefied  gas  made  luminous  by  an  electric  discharge,  or  that  from 
an  electric  spark  between  two  metallic  balls,  we  shall  get  a  "dis- 
continuous" spectrum  composed  of  numerous  distinct  bright  lines 
of  different  color  upon  a  dark  background. 

The  spectrum  of  sunlight  (either  direct  or  reflected)  is,  on  the  con- 
trary, characterized  by  a  multitude  of  dark  lines  crossing  a  brilliant 
background  of  continuous  spectrum  —  as  if  some  of  the  « fence  pick- 
ets '  had  been  destroyed,  leaving  gaps.  These  dark  lines  of  the  solar 
spectrum  are  known  as  the  "Fraunhofer  lines/'  because  first  studied 
and  mapped  by  him  in  1814,  though  some  of  them  had  been  noticed 
by  Wollaston  in -1801. 

313.  Integrating  and  Analyzing  Spectroscope.  —  If  we  simply 
direct  the  collimator  of  a  spectroscope  towards  a  distant  luminous 
object,  every  part  of  the  slit  receives  light  from  every  part  of  the 
object,  so  that  in  this  case  every  elementary  streak  of  the  spectrum 
is  a  spectrum  of  the  entire  body,  without  distinction  of  parts.  A 
spectroscope  used  in  this  way  is  said  to  be  an  integrating  instrument. 

If,  however,  we  interpose  a  lens  (the  object-glass  of  a  telescope) 
between  the  luminous  object  and  the  slit,  so  as  to  have  in  the  plane 
of  the  slit  a  distinct,  real  image  of  the  object,  then  the  top  of  the 
slit,  for  instance,  will  be  illuminated  wholly  by  light  from  one  part 
of  the  object,  the  middle  of  it  by  light  from  another  point,  and  the 
bottom  by  light  from  still  a  third.  The  spectrum  formed  by  the 
top  of  the  slit  belongs,  then,  to  the  light  from  that  particular  point 
of  the  object  whose  image  falls  upon  that  part  of  the  slit ;  and  so  of 
the  rest.  We  thus  separate  the  spectra  of  the  different  parts  of  the 
object,  and  so  optically  analyze  it.  An  instrument  thus  used  is 
spoken  of  as  an  "  analyzing  spectroscope"  The  combined  instrument 
formed  by  attaching  a  spectroscope  to  a  large  telescope  for  the 
spectroscopic  observation  of  the  heavenly  bodies  has  been  called  by 
Mr.  Lockyer l  a  "  telespectroscope." 

For  solar  purposes  a  grating  spectroscope  is  generally  better  than 
a  prismatic,  being  less  complicated  and  more  compact  for  a  given 
power. 

1  Now  Sir  Norman  Lockyer. 


212 


THE   SUN. 


Fig.  108  represents  the  large  grating  spectroscope  of  the  Halsted 
Observatory,  as  arranged  for  photography.  It  can  be  used  visually 
also  by  substituting  an  ordinary  view-telescope  for  the  photographic 
tube. 


FiQ.  108.  —  Large  Spectroscope  (fitted  for  photography).    By  permission  of 
D.  Appleton  &  Co. 


PRINCIPLES    OP    SPECTRUM   ANALYSIS.  213 

314.  Principles  upon  which  Spectrum  Analysis  Depends. — These, 
substantially  as  announced  by  Kirchhoff  in  1858,  but  with  some 
later  modifications,  are  the  three  following  :  — 

1.  A  continuous  spectrum  is  given  by  every  incandescent  body, 
the  molecules  of  which  so  interfere  with  each  other  as  to  prevent 
their  free,  independent,  luminous  vibration ;  that  is,  by  bodies  which 
are  either  solid  or  liquid,  or,  if  gaseous,  are  under  high  pressure. 

2.  The  spectrum  of  a  gaseous  element,  under  low  pressure,  is  dis- 
continuous, made  up  of  bright  lines,  often  hundreds  in  number,  and 
these  lines  are  characteristic  ;    that  is,  the  same  substance  under 
similar  conditions  always  gives  the  same  set  of  lines,  and  generally 
does  so  even  under  widely  different  conditions. 

3.  A  gaseous  substance  absorbs  from  white  light  passing  through 
it  precisely  those  rays  of  which  its  own  spectrum  consists.     The  spec- 
trum of  white  light  which  has  been  transmitted  through  it  then 
exhibits  a  "reversed"  spectrum  of  the  gas  ;  that  is,  one  which  shows 
dark  lines  instead  of  the  characteristic  bright  lines. 


Reversal  of  Spectrum 

U 


FIG.  109.  —  Reversal  of  the  Spectrum. 

Fig.  109  illustrates  this  principle.  Suppose  that  in  front  of  the 
slit  of  the  spectroscope  we  place  a  spirit  lamp  with  a  little  carbonate 
of  soda  and  some  salt  of  thallium  upon  the  wick.  We  shall  then 
get  a  spectrum  showing  the  two  yellow  lines  of  sodium  and  the  green 
line  of  thallium,  bright.  If  now  the  lime-light  be  started  right  behind 
the  lamp  flame,  we  shall  at  once  get  the  effect  shown  in  the  lower 


214  THE   SUN. 

figure,  —  a  continuous  spectrum  crossed  by  black  lines1  just  where 
the  bright  lines  were  before.  Insert  a  screen  between  the  lamp 
flame  and  the  lime,  and  the  dark  lines  instantly  show  bright  again. 
This  experiment  at  once  suggests  the  explanation  of  the  solar 
spectrum.  Its  bright  continuous  background  is  due  to  the  photo- 
spheric  clouds  which  act  the  part  of  the  lime-cylinder  in  the  experi- 
ment, and  the  dark  lines  are  produced  by  the  absorbing  action  of 
gases  and  vapors  which  lie  between  us  and  the  photosphere.  Some 
of  the  lines,  called  "telluric  lines,"  are  produced  in  our  own  atmos- 
phere, but  most  of  them  originate  close  to  the  solar  surface,  as  we 
shall  see.  (Art.  319.) 

315.  Chemical  Constituents  of  the  Sun.  —  By  taking  advantage  of 
these  principles  we  can  detect  the  presence  of  a  large  number  of  well- 
known  terrestrial  elements  in  the 
sun.  The  solar  spectrum  is 
crossed  by  dark  lines,  which, 
with  an  instrument  of  high  dis- 
persion, number  several  thou- 
sand, and  by  proper  arrangements 
it  is  possible  to  identify  among 
these  lines  many  which  are  due 
FIG.  no.  —  The  Comparison  Prism.  to  the  presence  in  the  sun's  lower 

atmosphere  of  known  terrestrial 

elements  in  the  state  of  vapor.  To  effect  the  comparison  necessary 
for  this  purpose,  the  spectroscope  must  be  so  arranged  that  the 
observer  can  have  before  him,  side  by  side,  the  spectrum  of  sunlight 
and  that  of  the  substance  to  be  tested.  In  order  to  do  this,  half  of 
the  slit  is  fitted  with  a  little  "comparison  prism"  so  called  (Fig.  110), 
which  reflects  into  it  the  light  from  tKe  sun,  while  the  other  half  of 
the  slit  receives  directly  the  light  of  some  flame  or  electric  spark. 
On  looking  into  the  eye-piece  of  the  spectroscope,  the  observer  will 
then  see  a  spectrum,  the  lower  half  of  which,  for  instance,  is  made 
by  sunlight,  while  the  upper  half  is  made  by  light  coming  from  an 
electric  aTc  or  spark  containing  the  vapor  of  the  metal  —  iron,  for 
instance. 

Photography  may  also  be  most  effectively  used  in  these  com- 
parisons instead  of  the  eye.      Fig.  Ill  is  a  rather  unsatisfactory 


1  Their  darkness  is  relative  only;  the  "black"  lines  are  actually  a  little 
brighter  than  before,  but  the  background  becomes  so  brilliant  that  they  appear 
dark  by  contrast. 


LIST    OF    ELEMENTS. 


215 


reproduction,  on  a  reduced  scale,  of  a  negative  recently  made  by 
Professor  Trowbridge  at  Cambridge.  The  lower  half  is  the  violet 
portion  of  the  spectrum  of  the  sun,  and  the  upper  half  that  of  the 
vapor  of  iron  in  an  electric  arc.  The  reader  can  see  for  himself 
with  what  absolute  certainty  such  a  photograph  indicates  the  pres- 
ence of  iron  in  the  solar  atmosphere.  A  few  of  the  lines  in  the 
photograph  which  do  not  show  corresponding  lines  in  the  solar 
spectrum  are  due  to  other  elements,  partly  impurities  in  the  carbons 
of  the  electric  arc. 


FlG.  111.  —  Comparison  of  the  Solar  Spectrum  with  that  of  Iron.    From  a  negative  by 
Professor  Trowbridge  with  a  concave-grating  spectroscope. 

316.  Elements  thus  far  Detected  in  the  Sun.  —  As  the  result  of 
such  comparisons  Eowland  in  1890  gave  the  following  list  of  36 
elements  whose  presence  in  the  sun  he  regarded  as  certainly  estab- 
lished, and  it  is  probable  that  a  few  more  will  ultimately  be  added. 
The  elements  are  arranged  according  to  the  intensity  of  the  dark 
lines  by  which  they  are  represented  in  the  solar  spectrum,  while  the 
appended  figures  denote  the  rank  which  each  would  hold  if  the 
arrangement  had  been  based  on  the  number  instead  of  the  intensity 
of  the  lines.  In  the  case  of  iron  the  number  exceeds  2000. 

An  asterisk  denotes  that  lines  of  the  element  indicated  appear 
often  or  always  as  bright  lines  in  the  spectrum  of  the  chromosphere 
(Art.  322). 

*Calcium,  11.  *Strontium,  23.  Copper,  30. 

*Iron,  1.  *  Vanadium,  8.  *Zinc,  29. 

*Hydrogen,  22.  *Barium,  24.  *Cadmium,  26. 

*Sodium,  20.  .    *Carbon,  7.  *Cerium,  10. 

*Mckel,  2.  Scandium,  12.  Glucinum,  33. 

*Magnesium,  19.  *Yttrium,  15.  Germanium,  32. 

*Cobalt,  6.  Zirconium,  9.  Rhodium,  27. 

Silicon,  21.  Molybdenum,  17.  Silver,  31. 

Aluminium,  25.  Lanthanum,  14.  Tin,  34. 

*Titanium,  3.  Niobium,  16.  Lead,  35. 

*Chromium,  5.  Palladium,  18.  Erbium,  28. 

*Manganese,  4.  Neodymium,  13.  Potassium,  36. 

To  these  must  now  be  added  helium,  which,  however,  shows  itself  only  by 
bright  lines  in  the  chromosphere  spectrum  (Art.  323). 


216  THE   SUN. 

317.  It  will  be  noticed  that  all  the  bodies  named  in  the  list, 
carbon  alone  excepted,  are  metals  (chemically  hydrogen  is  a  metal), 
and  that  a  multitude  of  the  most  important  terrestrial  elements  fail 
to  appear ;    oxygen,1  nitrogen,  chlorine,    bromine,  iodine,    sulphur, 
phosphorus,  arsenic,  and  boron  are  all  missing.     We  must  be  cau- 
tious, however,  as  to  negative  conclusions.     It  is  quite  conceivable 
that  the  spectra  of  these  bodies  under  solar  conditions  may  be  so 
different  from  their  spectra  as  presented  in  our  laboratories  that  we 
cannot  recognize  them ;  for  it  is  now  quite  certain  that  some  sub- 
stances, nitrogen,  for  instance,  under  different  conditions,  give  two 
or  more  widely  different  spectra. 

318.  Mr.  Lockyer's  Views.  —  Mr.  Lockyer  thinks  it  more  prob- 
able that  the  missing  substances  are  not  truly  elementary,  but  are 
decomposed  or  " dissociated "  on  the  sun  by  the  intense  heat,  and  so 
do  not  exist  there,  but  are  replaced  by  their  components  ;  he  believes, 
in  fact,  that  none  of  our  so-called  elements  are  really  elementary, 
but  that  all  are  decomposable,  and,  to  some  extent,  actually  decom- 
posed in  the  sun  and  stars,  and  some  of  them  by  the  electric  spark 
in  our  own  laboratories.     Granting  this,  a  crowd  of  interesting  and 
remarkable  spectroscopic  facts  find  easy  explanation.     At  the  same 
time  the  hypothesis  is  encumbered  with  great  difficulties  and  has 
not  yet  been  finally  accepted  by  physicists  and  chemists.     For  a 
full  statement  of  his  views  the  reader  is  referred  to  his  "Chemistry 
of  the  Sun." 

319.  The  Reversing  Layer.  —  According  to  Kirchhoff's  theory 
the  dark  lines  are  formed  by  the  passing  of  light  from  the  minute 
solid  and  liquid  particles  of  which  the  photospheric  clouds  are  sup- 
posed to  be  formed,  through  vapors  containing  the  substances  which 
we  recognize  in  the  solar  spectrum.     If  this  be  so,  the  spectrum  of 
the  gaseous  envelope,  which  by  its  absorption  forms  the  dark  lines, 
should  by  itself  show  a  spectrum  of  corresponding  bright  lines. 
The  opportunities  are  of  course  rare  when  it  is  possible  to  obtain 
the  spectrum  of  this  gas-stratum  alone  by  itself  ;  but  at  the  time  of 
a  total  eclipse,  at  the  moment  when  the  sun's  disc  has  just  been  ob- 
scured by  the  moon,  and  the  sun's  atmosphere  is  still  visible  beyond 
the  moon's  limb,  if  the  slit  of  the  spectroscope  be  carefully  adjusted 
to  the  proper  point,  the  observer  ought  to  see  this  bright-line  spec- 
trum.    The  author  succeeded  in  making  this  very  observation  at  the 
Spanish  eclipse  of  1870.     The  lines  of  the  solar  spectrum,  which  up 
to  the  final  obscuration  of  the  sun  had  remained  dark  as  usual  (with 
the  exception  of  a  few  belonging  to  the  spectrum  of  the  chromo- 

1  Ilunge  (1896)  found  evidence,  since  confirmed,  of-  the  presence  of  oxygen. 


SUN-SPOT    SPECTRUM. 


217 


sphere),  were  suddenly  "reversed/'  and  the  whole  length  of  the 
spectrum  was  filled  with  brilliant-colored  lines,  which  flashed  out 
quickly  and  then  gradually  faded  away,  disappearing  in  about  two 
seconds,  —  a  most  beautiful  thing  to  see.  Substantially  the  same 
thing  has  since  then  been  several  times  observed.1 

320.  The  natural  interpretation  of  this  phenomenon  is,  that  the  forma- 
tion of  the  dark  lines  in  the  solar  spectrum  is  mainly,  at  least,  produced  by  a  very 
thin  layer  close  down  to  the  photosphere,  since  the  moon's  motion  in  two  seconds 
would  cover  a  thickness  of  only  about  500  miles.     It  was  not  possible,  how- 
ever, to  be  certain  that  all  the  dark  lines  were  reversed,  and  in  this  uncer- 
tainty lies  the  possibility  of  a  different  interpretation.     Mr.  Lockyer  doubts 
the  existence  of  any  such  thin  stratum.1     According  to  his  views  the  solar 
atmosphere  is  very  extensive,  and  those  lines  of  iron,  which  correspond  to 
the  more  complex  combinations  of  its  constituents,  are  formed  only  in  the 
regions  of  lower  temperature,  high  up  in  the  sun's  atmosphere.     They  should 
appear  early  at  the  time  of  an  eclipse  and  last  long,  but  not  be  very  bright. 
Those  due  to  the  constituents  of  iron  which  are  found  only  close  down 
to  the  solar  surface  should  be  short  and  bright;  and  he  thinks  that  the 
numerous  bright  lines  observed  under  the  conditions  stated  are  due  to  such 
substances  only. 

321.  Sun-spot  Spectrum,  —  This  is  like  the  general  solar  spectrum, 
except  that  certain  lines  are  much  widened,  while  certain  others  are 
thinned,  and  sometimes  the  lines  of  hydrogen  and  a  few  other  sub- 
stances are  reversed  into  bright  lines ;  this  is  almost  always  the  case 
with  the  If  and  K  lines  of  calcium. 


TTh«  5I700' 


FIG.  111*.  —  Portion  of  Sun-spot  Spectrum,  from  Photograph  of  1893. 

Fig.  Ill*  is  from  a  photograph  of  the  yellow-green  portion  of  a 
sun-spot  spectrum  which  exhibits  the  principal  peculiarities  very 

1  During  the  eclipse  of  August,  1896,  Mr.  Shackleton,  in  Nova  Zembla,  ob- 
tained with  a  "  prismatic  camera  "  a  fine  photograph  of  the  spectrum  of  the  solar 
atmosphere  at  the  instant  after  totality  began.  The  picture  fully  confirms  the 
author's  visual  observations,  and  appears  to  establish  the  reality  of  the  "revers- 
ing layer."  The  bright  lines  of  the  chromosphere  spectrum  (Art.  322)  are  of 
course  specially  conspicuous,  but  all  the  prominent  Fraunhofer  lines  are  present, 
"  reversed  "  and  bright.  (See  also  note  at  end  of  Chapter  XI.) 


218 


THE    SUN. 


well.  The  central  dark  stripe  is  the  spectrum  of  the  nucleus  of  the 
spot,  the  fainter  stripe  on  each  side  of  it  being  that  of  the  penumbra. 
Rather  more  than  half  the  lines  are  unaltered  where  they  cross  the 
spot-spectrum,  about  twenty  (in  this  particular  spot  and  in  this 
piece  of  the  spectrum)  are  more  or  less  widened  and  darkened,  and 
about  half  a  dozen  are  thinned  or  obliterated.  Several  of  the  lines 
most  conspicuous  in  the  spot-spectrum  are  hardly  visible  at  all  in 
the  general  photospheric  spectrum,  the  two  "fish-bellies"  at  5728 
and  5731  being  specially  notable.  In  the  case  of  certain  elements, 
iron,  for  instance,  only  a  few  of  their  lines  are  ordinarily  affected 
in  the  spot-spectrum,  while  the  others  remain  unchanged,  —  a  fact 
which  Lockyer  considers  very  important  and  significant. 

Not   infrequently   it   happens 


FIG.  112.  —  The  C  Line  in  the  Spectrum  of  a 
Sun  Spot,  September  22, 1870. 


that  certain  lines  of  the  spectrum 
are  crooked  and  broken  in  con- 
nection with  sun  spots,  as  shown 
by  Fig.  112.  Such  phenomena 
are  caused,  according  to  Doppler's 
Principle,  by  the  swift  motion 
of  matter  towards  or  from  the 
observer.  In  the  particular  case 
shown  in  the  figure,  hydrogen  is  the  substance,  and  the  greatest 
motion  indicated  was  towards  the  observer  at  the  rate  of  about  300 
miles  a  second  —  an  unusual  velocity.  These  effects  are  most  notice- 
able, not  in  the  spots,  but  near  them,  usually  just  at  the  outer  edge 
of  the  penumbra. 

The  dark  and  apparently  continuous  spectrum  which  is  due  to  the  nucleus 
of  a  sun  spot  is  not  truly  continuous,  but  under  high  dispersion  is  resolved 
into  a  range  of  extremely  fine,  close-packed,  dark  lines,  separated  by  narrow 
spaces.  At  least,  this  is  so  in  the  green  and  blue  portions  of  the  spectrum; 
it  is  more  difficult  to  make  out  this  structure  in  the  yellow  and  red.  It  ap- 
pears to  indicate  that  the  absorbing  medium  which  causes  the  darkness  of  a 
sun  spot  is  gaseous,  and  not  composed  of  precipitated  particles  like  smoke. 

321*.  The  Doppler-Fizeau  Principle.  —  The  importance  of  this 
principle  in  the  "New  Astronomy"  can  hardly  be  exaggerated.  Briefly 
it  is  simply  this  :  When  the  distance  is  increasing  between  us  and  a 
body  which  is  emitting  regular  vibrations,  the  number  of  waves 
received  by  us  in  a  second  is  diminished,  and  their  wave-length  is 
correspondingly  increased;  and  vice  versa  when  the  distance  is  de- 
creasing. When,  therefore,  a  luminous  mass  (say  of  hydrogen)  is 
rushing  towards  us,  or  we  towards  it,  all  the  lines  in  its  spectrum 


CHROMOSPHERE   AND   PROMINENCES.  219 

have  their  wave-lengths  diminished,  and  are  shifted  from  their  nor- 
mal positions  in  the  spectrum  towards  the  blue  end,  by  an  amount 
depending  upon  the  velocity  of  the  motion. 

Doppler  first  announced  the  principle  in  1843  as  affecting  color; 
Fizeau  in  1848  pointed  out  its  effect  in  shifting  the  lines  of  the 
spectrum.  (See  also  Art.  802,*  first  fine-print  paragraph.) 

If  V  is  the  velocity  of  light  (186330  miles  a  second),  r  the  speed  with 
which  the  observer  is  moving  away  from  the  luminous  object,  and  s  the 
speed  with  which  the  object  is  moving  from  the  observer,  then,  letting  \ 
represent  the  normal  wave-length  of  a  given  line  in  the  spectrum,  and  \'  its 

V  +  s 
apparent  wave-length  as  affected  by  the  motions,  we  have  A/  =  A.  y  _   • 

If  r  and  s  are  both  small  as  compared  with  V,  as  of  course  is  usually  the 
case,  this  gives  very  approximately  A.'  —  A  —  A.  y  >  or  — - —  =  — ,  where  v 

is  simply  the  rate  at  which  the  distance  between  the  observer  and  the  object 
is  increasing  (to  be  taken  minus,  if  decreasing) .  With  our  present  spectro- 
scopes a  motion  of  less  than  a  mile  a  second  can  be  detected  in  this  way. 

322.,  The  Chromosphere.  —  The  chromosphere  is  a  region  of  the 
sun's  gaseous  envelope  which  lies  close  above  the  photosphere,  the 
"reversing  layer"  if  it  exists  at  all,  being  only  the  most  dense  and 
hottest  part  of  it.  The  chromosphere  is  so  called,  because,  as  seen 
for  an  instant,  during  a  total  solar  eclipse,  it  is  of  a  bright  scarlet 
color,  the  color  being  due  to  the  hydrogen  which  is  its  main  constit- 
uent. It  is  from  5000  to  10000  miles  in  thickness,  and  in  structure 
is  very  like  a  sheet  of  scarlet  flame,  not  being  composed  of  horizon- 
tal sheets,  but  of  (approximately)  upright  filaments.  Its  appearance 
has  been  compared  very  accurately  to  that  of  "a  prairie  on  fire"  ; 
but  the  student  must  carefully  guard  against  the  idea  that  there  is 
any  real  "  burning"  in  the  case;  i.e.,  any  process  of  combination  be- 
tween hydrogen  and  some  other  substance.  Its  spectrum  is  one  of 
bright  lines  containing  all  that  are  ever  observed  in  the  prominences, 
besides  many  others.  The  lines  of  hydrogen  and  helium,  and  the 
H  and  K  lines  of  calcium  are  the  most  conspicuous. 

323.  The  Prominences.  —  At  a  total  eclipse,  after  the  totality  has 
fairly  set  in,  there  are  usually  to  be  seen  at  the  edge  of  the  moon's 
disc  a  number  of  scarlet,  star-like  objects,  which  in  the  telescope 
appear  as  beautiful,  fiery  clouds  of  various  form  and  size.  These 
are  the  so-called  "prominences"  which  very  non-committal  name  was 
given  while  it  was  still  doubtful  whether  they  were  solar  or  lunar. 


220  THE   SUN. 

Photography,  in  1860,  proved  that  they  really  belong  to  the  sun, 
for  the  photographs  taken  during  the  totality  showed  that  the  moon 
obviously  moves  over  them,  covering  those  upon  the  eastern  limb, 
and  uncovering  those  upon  the  western. 

Their  spectrum,  first  observed  in  1868,  is  gaseous,1  i.e.,  bright-lined, 
the  lines  of  hydrogen  being  especially  conspicuous.  There  are, 
however,  a  number  of  other  bright  lines,  —  among  them  the  violet 
H  and  K  lines  ascribed  to  calcium,  and  a  yellow  line  known  as  DB 
because  it  is  near  the  two  D  lines  of  sodium.  This  is  due  to  "helium," 
first  identified  as  a  terrestrial  element  in  1895,  in  the  gas  liberated 
from  cleveite  (and  certain  other  rare  minerals). 

In  connection  with  this  eclipse,  Janssen,  who  observed  it  in  India, 
found  that  the  lines  of  the  prominence  spectrum  were  so  bright  that 
he  was  able  to  observe  them  the  next  day  after  the  eclipse  in  full 
sunlight ;  and  he  also  found  that  by  a  proper  management  of  his 
instrument  he  could  study  the  form  and  behavior  of  the  prominences 
nearly  as  well  without  an  eclipse  as  during  one.  Lockyer,  in  Eng- 
land, some  time  earlier  had  come  to  similar  conclusions  from  theo- 
retical grounds,  and  he  practically  perfected  his  discovery  a  few 
weeks  later  than  Janssen,  although  without  knowledge  of  what  he 
had  done.  By  a  remarkable  but  accidental  coincidence  their  discov- 
eries were  communicated  to  the  French  Academy  on  the  same  day; 
and  in  their  honor  the  French  have  struck  a  medal  bearing  their 
united  effigies. 

324,     How  the  Spectroscope  Makes  the  Prominences  Visible.  — 

The  only  reason  we  cannot  see  the  prominences  at  any  time  is  on 

account  of  the  bright  illumination  of  our 
own  atmosphere.  We  can  screen  off  the 
direct  light  of  the  sun  ;  but  we  cannot  screen 
off  the  reflected  sunlight  coming  from  the 
air  which  is  directly  between  us  and  the 
prominences  themselves ;  a  light  so  bril- 
liant that  the  prominences  cannot  be  seen 
through  it  without  some  kind  of  aid. 
FIG'113-  The  spectrum  of  this  air-light  is,  of 

Spectroscope  Slit  adjusted  for  , ,  , ,  3  , , 

observations  of  the  course,  just  the  same  as  that  of  the  sun, 

Prominences.  —  a  continuous  spectrum  with  the  same 

1  Tacchini  has  reported  the  existence  of  white  prominences  (giving  a  continu- 
ous Spectrum),  conspicuous  to  the  eye  and  on  the  photographs  in  the  eclipses  of 
1883  and  1889.  The  evidence,  however,  is  hardly  conclusive. 


DIFFERENT   KINDS    OF    PROMINENCES. 


221 


dark  lines  upon  it.  When,  therefore,  we  arrange  the  apparatus  as 
indicated  in  Fig.  113,  pointing  the  telescope  so  that  the  image  of 
the  sun's  limb  just  touches  the  slit  of  the  spectroscope,  then,  if  there 
is  a  prominence  at  that  point,  we  shall  have  in  our  spectroscope  two 
spectra  superposed  upon  each  other ;  namely,  the  spectrum  of  the 
air-illumination  and  that  of  the  prominence.  The  latter  is  a  spec- 
trum of  bright  lines,  or,  if  the  slit  is  opened  a  little,  of  bright  images 
of  whatever  part  of  the  prominence  may  fall  within  the  edges  of  the 
slit.  Now,  the  brightness  of  these  images  is  not  affected  by  any 
increase  of  dispersion  in  the  spectroscope.  Increase1  of  dispersion 
merely  sets  these  images  farther  apart,  without  making -them  fainter. 
The  spectrum  of  the  aerial  illumination,  on  the  other  hand,  is  made 
very  faint  by  its  extension ;  and,  moreover,  it  presents  dark  lines  (or 
spaces  when  the  slit  is  opened)  precisely  at  the  points  where  the 
bright  images  of  the  prominences  fall. 

A  spectroscope  of  dispersive  power  sufficient  to  divide  the  two  E 
lines,  attached  to  a  telescope  of  four  or  five  inches  aperture,  gives  a 
very  satisfactory  view  of  these 
beautiful  objects;  the  red  im- 
age corresponds  to  the  C  line, 
and  is  by  far  the  best  for  such 
observations,  though  the  Z>3 
line  or  the  F  line  can  also  be 
used.  When  the  instrument 
is  properly  adjusted,  the  slit 
opened  a  little,  and  the  image 
of  the  sun's  limb  brought  ex- 
actly to  the  edge  of  the  slit, 
the  observer  at  the  eye-piece 
of  the  spectroscope  will  see 
things  about  as  we  have  at- 
tempted to  represent  them  in 
Kg.  114,  as  if  he  were  looking 
at  the  clouds  in  an  evening 
sky  through  a  slightly  opened  window-blind.2  (See  also  Art.  326*.) 


FIG. 114. 

The  Chromosphere  and  Prominences  seen  in 
the  Spectroscope. 


325,     Different  Kinds  of  Prominences ;  Their  Forms  and  Motions. 

—  The  prominences  may  be  broadly  divided  into  two  classes,  —  the 

1  Too  high  dispersion  injures  the  definition,  however,  because  the  lines  in  the 
spectrum  of  hydrogen  are  rather  broad  and  hazy. 

2  The  observation  of  prominences  in  this  manner  was  first  effected  by  Huggins 
(now  Sir  William  Huggins)  in  1868. 


222 


THE   SUN. 


Clouds. 


Diffuse. 


Filamentary. 


Stemmed. 


Plumes. 


Horns. 


FIG.  115. 

Quiescent  Prominences.    Scale  75,000  Miles  to  the  Inch.    From  "  The  Sun,"  by  permission  of 

D.  Appleton  &  Co. 


ERUPTIVE    PROMINENCES. 


223 


Vertical  Filaments. 


Prominences,  Sept.  7, 1871, 12.30  P.M. 


Cyclone. 


Same  at  1.15  P.M. 


Flame.  Jets  near  Sun's  Limb,  Oct.  5, 1871. 

FIG.  116. 
Eruptive  Prominences.   From  "  The  Sun,"  by  permission  of  D.  Appleton  &  Co. 


224  THE   SUN. 

quiescent  or  diffused,  and  the  eruptive  or  "metallic,"  as  Secchi  calls 
them,  because  they  show  in  their  spectrum  the  lines  of  many  metals 
besides  hydrogen.  The  former,  illustrated  by  Fig.  115  (see  p.  222), 
are  immense  clouds,  often  60000  miles  in  height,  and  of  correspond- 
ing horizontal  dimensions,  either  resting  upon  the  chromosphere  or 
connected  with  it  by  slender  stems  like  great  banyan-trees.  They 
are  not  very  brilliant,  and  are  composed  almost  entirely  of  hydrogen 
and  "helium."  They  often  remain  nearly  unchanged  for  days  to- 
gether as  they  pass  over  the  sun's  limb.  They  are  found  on  all 
portions  of  the  disc,  at  the  poles  and  equator  as  well  as  in  the  spot 
zones.  Some  of  them  are  cloud-like  forms  floating  entirely  detached 
from  the  sun's  surface.  * 

Usually  these  clouds  are  simply  the  remnants  of  prominences 
which  appear  to  have  been  thrown  up  from  below,  but  in  some  cases 
they  actually  form  and  grow  larger  without  any  visible  connection 
with  the  chromosphere  —  a  fact  of  considerable  importance,  as 
showing  in  those  regions  the  presence  of  hydrogen,  invisible  to  our 
spectroscopes  until  somehow  or  other  it  is  made  to  give  out  the  rays 
of  its  familiar  spectrum.  All  the  forms  and  motions  of  the  promi- 
nences, it  may  be  said  further,  seem  to  indicate  the  same  thing  — 
that  they  exist  and  move,  not  in  a  vacuum,  but  in  a  medium 
of  density  comparable  with  their  own,  as  clouds  do  in  our  own 
atmosphere. 

326.  The  eruptive  prominences,  on  the  other  hand,  are  brilliant 
and  active,  not  usually  so  large  as  the  quiescent,  but  at  times  enor- 
mous, reaching  elevations  of  100000,  200000,  or  even  400000  miles. 
They  are  illustrated  by  Fig.  116.  Most  frequently  they  are  in  the 
form  of  spikes  or  flames ;  but  they  present  also  a  great  variety  of 
other  fantastic  shapes,  and  are  sometimes  so  brilliant  as  to  be  visible 
with  the  spectroscope  on  the  surface  of  the  sun  itself,  and  not  merely 
at  the  limb.  Generally  prominences  of  this  class  are  associated 
with  active  sun  spots,  while  both  classes  appear  to  be  connected 
with  the  faculse.  The  figures  given  are  from  drawings  of  individual 
prominences  that  have  been  observed  by  the  author  at  different  times. 

These  solar  clouds  are  most  fascinating  objects  to  watch,  on  ac- 
count of  the  beauty  of  their  forms,  and  the  rapidity  of  their  changes. 
In  the  case  of  the  eruptive  prominences  the  swiftness  of  the  changes 
is  sometimes  wonderful  —  portions  can  be  actually  seen  to  move, 
and  this  implies  a  real  velocity  of  at  least  250  miles  a  second,  so 
that  it  is  no  exaggeration  to  speak  of  such  phenomena  as  veritable 
"explosions";  of  course,  in  such  cases  the  lines  in  the  spectrum  are 


PHOTOGRAPHY  OF  PROMINENCES. 


225 


greatly  broken  and  distorted,  and  frequently  a  "  magnetic  storm " 
follows  upon  the  earth,  with  a  brilliant  Aurora  Borealis. 

The  number  visible  at  a  single  time  is  variable,  but  it  is  not  very 
unusual  to  find  as  many  as  twenty  on  the  sun's  limb  at  once. 

326*.  Photography  of  the  Prominences  and  Chromosphere. — With 
our  present  sensitive  dry  plates,  and  by  utilizing  the  H  and  K  lines, 
which,  like  the  hydrogen  lines,  are  always  reversed  in  the  spectrum 
of  the  chromosphere  and  prominences,  it  has  become  perfectly  easy 
to  photograph  these  objects  with  a  spectroscope  arranged  like  Fig. 
108.  Professor  George  E.  Hale,  of  Chicago,  and  Deslandres,  of 
Paris,  first  and  almost  simultaneously,  took  up  the  subject  in  1890, 
and  have  been  especially  successful.  Both  have  devised  ingenious 
arrangements  (called  spectro-heliographs)}  by  which  they  are  able  to 
obtain  pictures  of  the  chromosphere  and  prominences  around  the 
whole  circumference  of  the  sun  at  once. 


FiG.  116*.  —  Hand  K  in  Chromosphere  Spectrum.    By  permission  of  D.  Appleton  &  Co. 

It  should  be  noted  that,  while  in  the  ordinary  solar  spectrum  H  and  K 
are  broad,  hazy,  dark  lands  or  shades,  in  the  spectrum  of  a  prominence  they 
are  thin,  bright  lines  just  in  the  middle  of  the  dark  bands.  Moreover,  H  is 
double,  i.e.  there  is  a  strong,  bright  line  of  hydrogen  close  to  the  still  brighter 
calcium  line  (which  occupies  the  middle  of  the  band)  and  on  its  red-ward  side. 

Fig.  116*  is  from  a  photograph  of  the  chromosphere  spectrum,  made  by 
setting  the  slit  of  the  spectroscope  tangential  to  the  edge  of  the  sun.  The 
hydrogen  line  below  H  is  shown  and  also  a  second  hydrogen  line  above  K, 

1  See  Addendum  A,  following  page  580. 


226 


THE   SUN. 


marked  H£.  But  the  most  notable  feature  is  the  double  reversal  of  K,  H, 
and  its  hydrogen  companion  (which  is  known  as  He).  In  the  spectrum  of 
faculce,  H  and  K  are  usually  bright  and  doubly  reversed  in  the  same  man- 
ner as  in  the  chromosphere,  while  over  the  spots,  though  usually  reversed, 
they  are  seldom,  if  ever,  doubled. 

327.  The  Corona.  —  This  is  a  halo,  or  "glory,"  of  light  which 
surrounds  the  sun  at  the  time  of  the  total  eclipse.  From  the  remotest 
times  it  has  been  well  known,  and  described  with  enthusiasm,  as 
being  certainly  one  of  the  most  beautiful  of  natural  phenomena. 


FIG.  117.  —  Corona  of  the  Eclipse  of  1871.    By  permission  of  D.  Appleton  &  Co. 

The  portion  of  the  corona  nearest  the  sun  is  almost  dazzlingly 
bright,  with  a  greenish,  pearly  tinge  which  contrasts  finely  with  the 
scarlet  blaze  of  the  prominences.  It  is  made  up  of  streaks  and  fila- 
ments which  on  the  whole  radiate  outwards  from  the  sun's  disc, 
though  they  are  in  many  places  strangely  curved  and  intertwined. 
Usually  these  filaments  are  longest  in  the  sun-spot  zones,  thus  giv- 


THE   COKONA. 


227 


ing  the  corona  a  more  or  less  quadrangular  figure.  At  the  very  poles 
of  the  sun,  however,  there  are  often  tufts  of  sharply  defined  threads. 

For  the  most  part  the  streamers  have  a  length  not  much  exceeding 
the  sun's  radius,  but  some  of  them  at  almost  every  eclipse  go  far 
beyond  this  limit.  In  the  clear  air  of  Colorado  during  the  eclipse 
of  1878,  two  of  them  could  be  traced  for  five  or  six  degrees,  —  a  dis- 
tance of  at  least  9  000000  miles  from  the  sun.  A  most  striking 
feature  of  the  corona  usually  consists  of  certain  dark  rifts  which 
reach  straight  out  from  the  moon's  limb,  clear  to  the  extremest  limit 
of  the  corona. 

The  corona  varies  much  in  brightness  at  different  eclipses,  and  of 
course  the  details  are  never  twice  the  same.  Its  total  light  under 
ordinary  circumstances  is  at  least  two  or  three  times  as  great  as  that 
of  the  full  moon. 


328.  Photographs  of  the  Corona.  —  While  the  eye  can  perhaps 
grasp  some  of  its  details  more  satisfactorily  than  the  photographic 
plate  can  do,  it  is  found 
that  drawings  of  the  corona 
are  hardly  to  be  trusted. 
At  any  rate,  it  seldom  hap- 
pens that  the  representa- 
tions of  two  artists  agree 
sufficiently  to  justify  any 
confidence  in  their  scientific 
accuracy.  Photographs,  on 
the  other  hand,  may  be 
trusted  as  far  as  they  go, 
though  they  may  fail  to 
bring  out  some  things  which 
are  conspicuous  to  the  eye. 
Fig.  117  is  from  a  photo- 
graph of  the  Indian  eclipse 
of  1871,  and  117*  is  from 
the  photograph  of  the  Egyp- 
tian eclipse  of  1882,  when 
a  little  comet  was  found 

Close   to   the    SUn.  FIG.  117*.  —  Corona  of  the  Egyptian  Eclipse,  1882. 

Of  course,  as  in  the  case  of  the  prominences,  the  only  reason  we  cannot 
see  the  corona  without  an  eclipsed  sun  is  the  illumination  of  the  earth's  at- 
mosphere. If  we  could  ascend  above  our  atmosphere,  and  manage  to  exist 


228  THE   SUN. 

and  to  observe  there,  we  could  see  it  by  simply  screening  off  the  sun's  disc. 
So  long,  however,  as  the  brightness  of  the  illuminated  air  is  more  than 
about  sixty  times  that  of  the  corona,  it  must  remain  invisible  to  the  eye. 
Sir  William  Huggins  has  thought  that  it  might  be  possible  by  means  of 
photographs  to  detect  differences  of  illumination  less  than  ^L  (the  limit  of 
the  eye's  perception),  and  so  to  obtain  pictures  of  the  corona  at  any  time ; 
especially  as  it  appears  that  the  coronal  light  is  far  richer  in  ultra-violet 
rays  (the  photographic  rays)  than  the  general  sunlight  with  which  the  air  is 
illuminated.  Thus  far,  however,  no  success  has  been  obtained  either  by 
himself  or  by  others  who  have  made  the  attempt. 

329.  Spectrum  of  the  Corona.  —  This  was  first  definitely  observed 
in  1869  during  the  eclipse  which  passed  over  the  western  part  of  the 
United  States  in  that  year.  It  was  then  found  that  its  most  remark- 
able characteristic  is  a  bright  line  in  the  green,  which  the  writer 
incorrectly  identified  as  coinciding  with  the  dark  line  at  1474  on  the 
scale  of  KirchhofFs  map  (X  =  5317).  This  line  was  also  seen  by 
two  or  three  other  observers,  but  either  not  recognized  as  belonging 
to  the  corona,  or  differently  identified. 

This  result  was  very  puzzling,  since  the  line,  also  conspicuous  in  the 
chromosphere  spectrum,  is  ascribed  to  iron  by  Angstrom  and  other  authori- 
ties. The  mystery  was  cleared  up  by  the  eclipse-photographs1  obtained 
since  1896,  which  show  that  the  real  corona  line  is  not  "  1474  "  at  all,  but 
is  slightly  more  refrangible  with  a  wave-length  of  5304,  and  has  no  corre- 
sponding line  either  in  the  ordinary  solar  spectrum  or  in  that  of  the 
chromosphere.  It  is  now  generally  referred  to  an  hypothetical  element 
provisionally  named  coronium,  lighter  than  hydrogen,  existing  upon  the 
earth,  like  helium,  only  sparingly  if  at  all,  and  as  yet  unrecognized  in  our 
laboratories. 

Besides  this  conspicuous  green  line,  the  hydrogen  lines  are  also 
faintly  visible  in  the  spectrum  of  the  corona ;  and  by  means  of  a 
photographic  camera  used  during  the  Egyptian  eclipse  of  1882,  it 
was  found  that  the  upper  or  violet  portion  of  the  spectrum  is  very 
rich  in  lines,  among  which  If  and  K  were  specially  conspicuous. 
Later  observations,  however,  in  1893  and  1896,  have  made  it  nearly 
certain  that  these  lines  were  not  really  coronal,  but  only  due  to 
reflection  in  our  own  atmosphere  of  light  from  the  chromosphere 
and  prominences.  There  is  also,  through  the  whole  spectrum,  a 
faint  continuous  background.  In  it  some  observers  have  reported  the 
presence  of  a  few  of  the  more  conspicuous  dark  lines  of  the  ordinary 
solar  spectrum,  but  the  evidence  on  this  point  is  rather  conflicting. 

When  the  corona  is  photographed  with  a  " prismatic  camera," 
which  has  a  prism  or  prisms  in  front  of  its  lens,  the  picture  is  com- 

1  See  notes  on  pages  231  and  267. 


NATURE    OF   THE    CORONA.  229 

posed  of  several  rings  (seven  in  1893),  all  of  which,  except  the 
green  one,  are  very  faint  and  lie  in  the  violet  portion  of  the  spec- 
trum; the  brightest  of  them  falls  just  below  the  IT  line.  They  are 
all  probably  due  to  "coroniuin."  The  plate  at  the  same  time  also 
shows  numerous  other  partial  rings  which  are  quite  different  in 
appearance,  and  are  clearly  due  to  the  hydrogen,  helium,  and  calcium 
of  the  prominences.  As  the  evidence  now  stands,  it  is  not  probable 
that  either  of  these  elements  exists  in  the  corona. 

330.  Nature  of  the  Corona,  —  It  is  evident  that  the  corona  is  a 
truly  solar  and  not  merely  an  optical  or  atmospheric  phenomenon, 
from  two  facts  :  first,  the  identity  of  detail  in  photographs  made  at 
widely  separate  stations.     In  1871,  for  instance,  photographs  were 
obtained  at  the  Indian  station  of  Bekul,  in  Ceylon,  and  in  Java, 
three  stations  separated  by  many  hundreds  of  miles  ;  but,  excepting 
minute  differences  of  detail,  such  as  might  be  expected  to  have 
resulted  from  the  changes  that  would  naturally  go  on  in  the  corona 
during  the  half-hour  while  the  moon's  shadow  was  travelling  from 
Bekul  to  Java,  all  the  photographs  agree  exactly,  which  of  course 
would  not  be  the  case  if  the  corona  depended  in  any  way  upon  the 
atmospheric  conditions  at  the  observer's  station. 

Second  (but  first  historically),  the  presence  of  bright  lines  in  the 
spectrum  of  the  corona  proves  that  it  cannot  be  a  terrestrial  or  lunar 
phenomenon,  by  demonstrating  the  presence  in  the  corona  of  a  self- 
luminous  gas,  which  observation  fails  to  find  either  near  to  the  moon 
or  in  our  own  atmosphere.  It  must,  therefore,  be  at  the  sun. 

But  while  it  is  thus  certain  that  the  corona  contains  luminous  gas, 
it  also  is  very  likely  that  finely  divided  solid  or  liquid  matter  may 
be  present  in  the  corona ;  that  is,  fog  or  dust  of  some  kind,  as  is 
indicated  by  the  partial  polarization  of  its  light. 

331.  The  corona  cannot  be  a  true  "solar  atmosphere"  in  any 
strict  sense  of  the  word.     No  gaseous  envelope  in  any  way  analogous 
to  the  earth's  atmosphere  could  possibly  exist  there  in  gravitational 
equilibrium  under  the  solar  conditions  of  pressure  and  temperature. 
The  corona  is  probably  a  phenomenon  due  somehow  to  the  intense 
activity  of  the  forces  there  at  work ;   meteoric  matter,  cometic  mat- 
ter, matter  ejected  from  within  the  sun,  are  all  concerned. 

That  this  matter  is  inconceivably  rare  is  evident  from  the  fact 
that  in  several  cases  comets  have  passed  directly  through  the  corona 
without  experiencing  the  least  perceptible  disturbance  of  their  mo- 


280  THE   SUN. 

tions.  It  is  altogether  probable  that  at  a  very  few  thousand  miles 
above  the  sun's  surface  its  density  becomes  far  less  than  that  of  the 
best  vacuum  we  can  make  in  an  electric  lamp. 

No  wholly  satisfactory  theory  of  the  corona  has  yet  been  found. 
Before  1869  it  was  very  generally  regarded  as  a  purely  optical  phe- 
nomenon, either  due  to  diffraction  at  the  limb  of  the  moon,  or,  like 
rainbows  and  halos,  produced  in  our  own  atmosphere.  Later,  some 
sought  an  explanation  in  meteoric  matter  descending  upon  the  sun 
from  interplanetary  space.  Many,  recognizing  the  striking  resem- 
blance between  the  appearance  of  the  corona  and  that  of  the  Aurora 
Borealis,  have  inferred  a  similarity  of  nature  ;  that  the  corona,  in 
short,  is  a  "permanent  solar  aurora/7  consisting  of  streams  of  elec- 
trical discharge,  directed  and  arranged  by  solar  magnetism.  A 
" magnetic  theory"  based  on  this  general  idea  has  been  elaborately 
developed  by  Professor  Bigelow,  of  the  U.  S.  Weather  Bureau. 

Professor  Schaeberle,  of  the  Lick  Observatory,  on  the  other  hand, 
contends  for  a  purely  "mechanical"  theory,  regarding  the  coronal 
streamers  as  jets  of  rare  material  ejected  from  the  solar  surface 
(chiefly  from  the  sun-spot  zones)  to  planetary  distances,  from  which 
it  falls  back  in  a  state  of  diffusion.  To  this  returning  matter  he 
attributes  many  important  solar  phenomena. 

None  of  the  theories,  however,  seem  to  allow  sufficient  weight  to 
the  fundamental  spectroscopic  fact  that  "coronium,"  the  character- 
istic substance  of  the  corona,  appears  thus  far  to  be  absolutely  unique 
in  nature,  —  utterly  distinct  from  any  other  known  form  of  matter, 
terrestrial,  solar,  or  cosmical. 


EXERCISES  ON  CHAPTERS  VIII  AND  IX. 

1.  Assuming  Faye's  equation  for  the  solar  rotation  (Art.  284),  what  are 
the  rotation-periods  at  the  sun's  equator,  in  latitude  30°,  in  latitude  45°, 
and  at  the  pole  ? 

(At  equator,  25.06  days. 
Lat.  30°,     26.49     « 
Lat.  45°,     28.09     « 
At  pole,      31.95     « 

2.  What  would  be  the  synodic  or  apparent  time  of  rotation  for  a  spot 
in  latitude  45°? 

Ans.   30.43  days. 


EXERCISES.  231 

3.  If  the  diameter  of  the   sun  were  doubled,  its  density  remaining 
unchanged,  what  would  be  the  force  of  gravity  at  its  surface  ? 

4.  If  the  sun  were  expanded  into  a  homogeneous  sphere,  with  a  radius 
equal  to  the  distance  from  the  earth  to  the  sun,  its  mass  remaining  un- 
changed, what  would  be  the  force  of  gravity  at  its  surface  ? 

5.  In  this  case,  what  change,  if  any,  would  result  in  the  orbit  of  the 
earth? 

6.  In  the  neighborhood  of  a  sun  spot  a  point  is  found  in  its  spectrum 
where  a  portion  of  the  C  line  (A  =  6563.0)  is  deflected  to  6566.0.     What 
is  the  velocity  (in  the  line  of  sight)  of  the  hydrogen  at  that  point  ?     (See 
Art.  321*.) 

Ans.    85.17  miles,  receding. 

7.  How  great  is  the  displacement  of  the  hydrogen  line  F  (A.  —  4861.5) 

at  that  point  ? 

Ans.    2.22  units  (of  wave-length). 

8.  How  great  a  displacement  is  produced  in  the  line  D  (\  =  5896.16) 
by  a  velocity  of  100  miles  a  second  ? 

Ans.    3.16  units. 

9.  If  a  luminous  body  were  moving  towards  us  with  a  velocity  one 
quarter  that  of  light,  what  would  be  the  effect  upon  the  apparent  length  of 
the  portion  of  the  spectrum  included  between  two  lines,  —  say  C  and  F1 

10.  What  if  it  were  moving  towards  us  with  the  speed  of  light  ? 

11.  What  if  it  were  receding  at  that  rate? 

Ans.    The  wave-length  of  every  ray  would  be  apparently  doubled. 

NOTE  TO  ART.  329. 

THE  COKONA  LINE.  The  photographs  of  spectra  obtained  in  India  by 
Lockyer  and  Campbell  during  the  eclipse  of  January,  1899,  made  it  very 
probable  that  the  true  wave-length  of  the  corona  line  is  5304,  instead  of  5316. 
The  "  1474  "  line  (X  =  5316)  is  by  far  the  most  conspicuous  of  the  chromosphere 
lines  in  that  region  of  the  spectrum,  persisting  some  time  after  the  others  vanish, 
when  the  total  phase  of  the  eclipse  begins,  and  it  would  be  easy  to  make  an 
erroneous  identification  of  the  corona  line  with  it.  It  now  seems  certain  that 
this  mistake  was  actually  made  in  1869.  The  results  obtained  in  the  eclipses  of 
May,  1900  and  1901,  have  confirmed  those  of  the  Indian  eclipse. 


232 


THE   SUN. 


CHAPTER   X. 

THE  SUN'S  LIGHT  AND  HEAT :  COMPARISON  OF  SUNLIGHT  WITH 
ARTIFICIAL  LIGHTS.  —  MEASUREMENT  OF  THE  SUN'S  HEAT, 
AND  DETERMINATION  OF  THE  "SOLAR  CONSTANT." — PYR- 
HELIOMETER,  ACTINOMETER,  AND  BOLOMETER. — THE  SUN'S 
TEMPERATURE.  —  THEORIES  AS  TO  THE  MAINTENANCE  OF 
THE  SUN'S  RADIATION,  AND  CONCLUSIONS  AS  TO  THE  SUN?S 
POSSIBLE  AGE  AND  FUTURE  DURATION. 

332.  The  Sun's  Light.  —  The  Quantity  of  Sunlight.    It  is  very  easy 
to  compare   (approximately)  sunlight  with  the  light  of  a  standard1 
candle ;  and  the  result  is,  that  when  the  sun  is  in  the  zenith,  it  illumi- 
nates a  white  surface  about  60,000  times  as  strongly  as  a  standard 
candle  at  a  distance  of  one  metre.    If  we  allow  for  the  atmospheric  ab- 
sorption, the  number  would  be  fully  70,000.   If  we  then  multiply  70,000 
by  the  square  of  150,000  million  (roughly  the  number  of  metres  in 
the  sun's  distance  from  the  earth) ,  we  shall  get  what  a  gas  engineer 
would  call  the  sun's  "candle  power."     The  number  comes  out  1575 
billions   of   billions  (English);   i.e.,  1575  with  twenty-four  ciphers 
following. 

333.  One  way  of  making  the  comparison  is  the  following :  Arrange  mat- 
ters as  in  Fig.  118.     The  sunlight  is  brought  into  a  darkened  room  by  a 
mirror  M,  which  reflects  the  rays  through  a  lens  L  of  perhaps  half  an  inch  in 
diameter.     After  the  rays  pass  the   focus  they  diverge   and   form  on  the 
screen  S  a  disc  of  light,  the  size  of  which  may  be  varied  by  changing  the 
distance  of  the  screen.     Suppose  it  so  placed  that  the  illuminated  circle  is 
just  ten  feet  in  diameter;  that  is,  240  times  the  diameter  of  the  lens.     The 
illumination  of  the  disc  will  then  be  less  than  that  of  direct  sunlight  in 
the  ratio  of  2402  (or  57,600)  to  1  (neglecting  the  loss  of  light  produced  by 

i  A  standard  candle  is  a  sperm  candle  weighing  one-sixth  of  a  pound  and  burn- 
ing 120  grains  an  hour.  The  "decimal  candle"  proposed  by  the  Paris  Congress 
in  1890  is  about  one  per  cent  less.  It  is  one-twentieth  of  the  light  emitted  by  a 
square  centimetre  of  melting  platinum.  An  ordinary  gas-burner  consuming  five 
feet  of  gas  hourly  gives  a  light  equivalent  to  from  twelve  to  fifteen  standard  candles. 


INTENSITY    OF    THE    SUN  S    LUMINOSITY, 


233 


the  mirr.or  and  the  lens,  a  loss  which  of  course  must  be  allowed  for).  Now 
place  a  little  rod  like  a  pencil  near  the  screen,  as  at  P,  light  a  standard 
candle,  and  move  the  candle  back  and  forth  until  the  two  shadows  of  the 
pencil,  one  formed  by  the  candle,  and  the  other  by  the  light  from  the  lens, 


FIG.  118.  —  Comparison  of  Sunlight  with  a  Standard  Candle. 

are  equally  dark.  It  will  be  found  that  the  candle  has  to  be  put  at  a  dis- 
tance of  about  one  metre  from  the  screen  ;  though  the  results  would  vary  a 
good  deal  from  day  to  day  with  the  clearness  of  the  air. 

334.  When  the  sun's  light  is  compared  with  that  of  the  full  moon 
and  of  various  stars,  we  find,  as  stated  (Art.  259),  that  it  is  about 
600000  times  that  of  the  full  moon.  It  is  7000  000000  times  as 
great  as  the  light  received  from  Sirius,  and  about  40000  000000 
times  that  from  Vega  or  Arcturus. 


335.  The  Intensity  of  the  Sun's  Luminosity.  —  This  is  a  very 
different  thing  from  the  total  quantity  of  its  light,  as  expressed  by 
its  "candle  power77  (a  surface  of  comparatively  feeble  luminosity 
can  give  a  great  quantity  of  light  if  large  enough).  It  is  the  amount 
of  light  per  square  inch  of  luminous  surface  which  determines  the 
intensity.  Making  the  necessary  computations  from  the  best  data 
obtainable  (only  roughish  approximations  being  possible),  it  appears 
that  the  sun's  surface  is  about  190000  times  as  bright  as  that  of  a 


284 


THE  stnsr. 


candle  flame,  and  about  150  times  as  bright  as  the  lime  of  the  calcium 
light.  Even  the  darkest  part  of  a  solar  spot  outshines  the  lime.  The 
intensely  brilliant  spot  in  the  so-called  "crater"  of  an  electric  arc 
comes  nearer  sunlight  than  anything  else  known,  being  from  one-half 
to  one-fourth  as  bright  as  the  surface  of  the  sun  itself.  But  either 
the  electric  arc  or  the  calcium  light,  when  interposed  between  the 
eye  and  the  sun  looks  like  a  dark  spot  on  the  disc. 

336.  Comparative  Brightness  of  Different  Portions  of  the  Sun's 
Surface.  — By  forming  a  large  image  of  the  sun,  say  a  foot  in  di- 
ameter, upon  a  screen,  we  can  compare  with  each  other  the  rays 
coming  from  different  parts  of  the  sun's  disc.     It  thus  appears  that 
there  is  a  great  diminution  of  light  at  the  edge,  the  light  there,  accord- 
ing to  Professor  Pickering's  experiments,  being  just  about  one-third 
as  strong  as  at  the  centre.     There  is  also  an  obvious  difference  of 
color,  the  light  from  the  edge  of  the  disc  being  brownish  red  as  com- 
pared with  that  from  the  centre.     The  reason  is,  that  the  red  and 
yellow  rays  of  the  spectrum  lose  much  less  of  their  brightness  at  the 
limb  than  do  the  blue  and  violet.    According  to  Vogel,  the  latter  rays 
are  affected  nearly  twice  as  much  as  the  former.     For  this  reason, 
photographs  ot  the  sun  exhibit  the  darkening  of  the  limb  much  more 
strongly  than  one  usually  sees  it  in  the  telescope. 

337.  Cause  of  the  Darkening  of  the  Limb.  —  It  is  due  unques- 
tionably to  the  general  absorption  of  the  sun's  rays  by  the  lower  por- 
tion of  the  overlying  atmosphere. 
The    reason    is  obvious   from   the 
figure  ( Fig.  119).     The  thinner  this 
atmosphere,    other    things     being 
equal,  the  greater  the  ratio  between 
the  percentage  of  absorption  at  the 
edge  and  centre  of  the  disc,  and 
the  more  obvious  the  darkening  of 
the  limb. 

Attempts    have    been    made    to 
determine  from  the   observed  dif- 
brightness    of    centre    and    limb   the    total 
Unfortunately  we  have 


FIG.  119. 
Cause  of  the  Darkening  of  the  Sun's  Limb. 


ferences    between    the 

percentage  of  the  sun's  light  thus  absorbed. 
to  supplement  the  observed  data  with  some  very  uncertain  assump- 
tions in  order  to  solve  the  problem ;  and  it  can  only  be  said  that 
it  is  probable  that  the  amount  of  light  absorbed  by  the  sun's  atmos- 


THE   SUN'S    HEAT.  235 

phere  lies  between  fifty  and  eighty  per  cent ;  i.e.,  the  sun  deprived 
of  its  gaseous  envelope  would  probably  shine  from  two  to  five  times 
as  brightly  as  now.  It  is  noticeable,  also,  as  Langley  long  ago 
pointed  out,  that  thus  stripped,  the  "complexion"  of  the  sun  would 
be  markedly  changed  from  yellowish  white  to  a  good  full  blue,  since 
the  blue  and  violet  rays  are  much  more  powerfully  absorbed  than 
those  at  the  lower  end  of  the  spectrum. 


THE  SUN'S  HEAT. 

338.  Its  Quantity ;  the  "  Solar  Constant."  —  By  the  "quantity  of 
heat "  received  by  the  earth  from  the  sun  we  mean  the  number  of 
heat-units  received  in  each  unit  of  time  by  a  square  unit  of  surface 
when  the  sun  is  in  the  zenith.     The  heat-unit  most  employed  by 
engineers  is  the  calorie,  which  is  the  quantity  of  heat  required  to 
raise  the  temperature  of  one  kilogram  of  water  one  degree  centi- 
grade.    It  is  found  by  observation  that  each  square  metre  of  surface 
exposed  perpendicularly  to  the  sun's  rays  receives  from  the  sun  each 
minute  approximately  twenty-one  of  these  calories  ;   or  rather  it 
would  do  so  if  a  considerable  portion  of  the  sun's  heat  were  not 
stopped  by  the  earth's  atmosphere,  which  absorbs  some  thirty  per 
cent  of  the  whole,  even  when  the  sun  is  vertical,  and  a  much  larger 
proportion  when  the   sun   is  near   the   horizon.      This    quantity, 
twenty-one  calories l  per  square  metre  per  minute,  is  known  as  the 
"Solar  Constant." 

339.  Method  of  Determining  the  "  Solar  Constant."—  The  method 
by  which  the  solar  constant  is  determined  is  simple  enough  in  prin- 
ciple, though  complicated  with  serious  practical  difficulties  which 
affect  its  accuracy.     It  is  done  by  allowing  a  beam  of  sunlight  of 
known  cross-section  to  shine  upon  a  known  weight  of  water  (or  other 
substance  of  known  specific  heat)  for  a  known  length  of  time,  and 

1  For  many  scientific  purposes  the  engineering  calorie  is  inconveniently  large, 
and  a  smaller  one  is  employed,  which  replaces  the  kilogram  of  water  by  the 
gram  heated  one  degree  —  the  smaller  calorie  being  thus  only  T^  of  the  engi- 
neering unit.  As  stated  by  many  writers  (Langley,  for  instance),  the  solar  con- 
stant is  the  number  of  these  small  calories  received  per  square  centimetre  of 
surface  in  a  minute.  This  would  make  the  number  2.1  instead  of  21.  It  would 
perhaps  be  better  to  bring  the  whole  down  to  the  "c.g.s.  system"  by  substitut- 
ing the  second  for  the  minute ;  and  this  would  give  us  for  the  solar  constant 
about  one-thirtieth  of  a  (small)  calorie  per  square  centimetre  per  second. 


236 


THE   SUN. 


measuring  the  rise  of  temperature.  It  is  necessary,  however,  to  de- 
termine and  allow  for  the  heat  received  from  other  sources  during  the 
experiment,  and  for  that  lost  by  radiation.  Above  all,  the  absorb- 
ing effect  of  our  own  atmosphere  is  to  be  taken  into  account,  and 
this  is  the  most  difficult  and  uncertain  part  of  the  work,  since  the 
atmospheric  absorption  is  continually  changing  with  every  change  of 
the  transparency  of  the  air,  or  of  the  sun's  altitude. 

340.    Pyrheliometers    and   Actinometers.  —  The   instruments  with 
which  these  measurements  are  made,  are  known  as  "  pyrheliometers  "  and 

"  actinometers."  Fig.  120  represents  the  pyr- 
heliometer  of  Pouillet,  with  which  in  1838  he 
made  his  determination  of  the  solar  constant, 
at  the  same  time  that  Sir  John  Herschel  was 
experimenting  at  the  Cape  of  Good  Hope  in 
practically  the  same  way.  They  were  the 
first  apparently  to  understand  and  attack  the 
problem  in  a  reasonable  manner.  The  pyr- 
heliometer  consists  essentially  of  a  little  cylin- 
drical box  ab,  like  a  snuff-box,  made  of  thin 
silver  plate,  with  a  diameter  of  one  decimetre 
and  such  a  thickness  that  it  holds  100  grams 
of  water.  The  upper  surface  is  carefully 
blackened,  while  the  rest  is  polished  as  brill- 
iantly as  possible.  In  the  water  is  inserted 
the  bulb  of  a  delicate  thermometer,  and  the 
whole  is  so  mounted  that  it  can  be  turned  in 
any  direction  so  as  to  point  it  directly  towards 
the  sun.  It  is  used  by  first  holding  a  screen 
between  it  and  the  sun  for  (say)  five  minutes, 
and  watching  the  rise  or  fall  of  the  mercury  in 
the  thermometer  at  m.  There  will  usually  be 
some  slight  change  due  to  the  radiation  of 
surrounding  bodies.  The  screen  is  then  re- 
moved, and  the  sun  is  allowed  to  shine  upon 

FiG.l20.-Pouillet'sPyrheliometer.  the  blackened  surface   for  five  minutes,  the 

instrument   being    continually   turned    upon 

the  thermometer  as  an  axis,  in  order  to  keep  the  water  in  the  calorimeter 
box  well  stirred.  At  the  end  of  the  five  minutes  the  screen  is  replaced 
and  the  rise  of  the  temperature  noted.  The  difference  between  this  and 
the  change  of  the  thermometer  during  the  first  five  minutes  will  give  us  the 
amount  by  which  a  beam  of  sunlight  one  decimetre  in  diameter  has  raised 
the  temperature  of  100  grams  of  water  in  five  minutes,  and  were  it  not  for 
the  troublesome  corrections  which  must  be  made,  would  furnish  directly  the 
value  of  the  solar  constant. 


CORRECTION    FOR    ATMOSPHERIC    ABSORPTION. 


237 


341 .  The  second  apparatus,  Fig.  121,  is  the  actinometer  of  Violle,  which 
consists  of  two  concentric  metal  spheres,  the  inner  of  which  is  blackened  on 
the  inside,  while  the  outer  one  is  brightly  polished,  the  space  between  the 
two  being  filled  with  water  at  a  known  temperature,  kept  circulating  by  a 
pump  of  some  kind.  The  thermoscopic 
body  in  this  case,  instead  of  being  a  box 
filled  with  water,  is  the  blackened  bulb 
of  the  thermometer  T;  and  the  obser- 
vations may  be  made  either  in  the  same 
way  as  with  the  pyrheliometer,  or  simply 
by  noting  the  difference  between  the 
temperature  finally  attained  by  the  ther- 
mometer T  after  it  has  ceased  to  rise  in 
the  sun's  rays,  and  the  temperature  of 
the  water  circulating  in  the  shell. 


342.  Correction  for  Atmospheric 
Absorption.  —  The  correction  for  at- 
mospheric absorption  is  determined  by 
making  observations  at  various  altitudes 
of  the  sun  between  zenith  and  horizon. 
If  the  rays  were  homogeneous  (that  is, 
all  of  one  wave-length),  it  would  be 

comparatively  easy  to  deduce  the  true  correction  and  the  true  value  of 
the  solar  constant.  In  fact,  however,  the  visible  solar  spectrum  is  but  a 
small  portion  of  the  whole  spectrum  of  the  sun's  radiance,  and,  as  Langley 
has  shown,  it  is  necessary  to  determine  the  coefficient  of  absorption  separately 
for  all  the  rays  of  different  wave-length. 


FIG.  121.  — Violle's  Actinometer. 


343.  The  Bolometer.  —  This  he  has  done  by  means  of  his  "  Bolometer," 
an  instrument  which  is  capable  of  indicating  exceedingly  minute  changes  in 
the  amount  of  radiation  received  by  an  extremely  thin  strip  of  metal.  This 
strip  is  so  arranged  that  the  least  change  in  its  electrical  resistance  due  to 
any  change  of  temperature  will  disturb  a  delicate  galvanometer.  The 
instrument  is  far  more  sensitive  than  any  thermometer  or  even  thermo- 
pile, and  has  the  especial  advantage  of  being  extremely  quick  in  its  re- 
sponse to  any  change  of  radiation.  Fig.  122  shows  it  so  connected  with 
a  spectroscope  that  the  observer  can  bring  to  the  bolometer,  B,  rays  of 
any  wave-length  he  chooses.  The  rays  enter  through  the  collimator  lens 
L,  and  are  then  refracted  by  the  rock-salt  prism  P  (or  diffracted  by  a 
grating)  to  the  reflector  M,  whence  they  are  sent  back  to  B,  and  thus  pro- 
duce their  electrical  effect,  which  is  transmitted  to  the  galvanometer.  As 
the  galvanometer  needle  swings  one  way  or  the  other,  a  pencil  of  light 
reflected  from  it  falls  upon  a  sensitive  photographic  plate  which  is  moved 
by  the  same  clockwork  (not  shown  in  the  figure)  which  moves  the  prismj 


238 


THE   SUN. 


and  as  a  consequence  the  spot  of  light  traces  out  upon  the  plate  an  irregular 
curve,  in  which  the  "  hills  "  correspond  to  rise  of  temperature,  and  the  "  val- 
leys "  to  cooler  places  in  the  spec- 
trum. The  curve  may  then  be 
transformed  into  a  spectrum-map 
of  the  usual  form,  —  a  "holo- 
graph," as  it  is  called.  Fig.  122* 
is  from  Langley's  paper  of  1894. 
Langley  considered  the  atmos- 
pheric corrections  applied  by  the 
earlier  observers  too  low,  and  for 
some  years  the  adopted  value  of 
the  solar  constant  was  from  25  to 
30.  Scheiner  (1899)  considered 
40  as  more  probable,  but  on  the 
other  hand  the  Smithsonian  ob- 
servations of  1902-1903,  as  also 
those  of  1907,  give  from  19  to 
21.  It  may  be  found  that  the 
supposed  "constant"  is  widely 
variable. 

344.    A  less  technical  state- 
FlG- 122-  ment  of   the  solar  radiation 

Langley's  Spectro-Bolometer ,  as  used  for  Mapping     rna^vkArnQrlp;T1  +PTMT,  <?  nf  fV,  i  PV 
the  Energy  of  the  Prismatic  Spectrum.  mW  D 

ness   of   the  quantity  of  ice 

which  would  be  melted  by  it  in  a  given  time.  Since  it  requires  79J 
calories  to  melt  a  kilogram  of  ice  with  a  density  of  0.92,  it  follows 
that  21  calories  a  minute  would  melt  in  an  hour  a  sheet  of  ice  one 
metre  square  and  17.3  millimetres  (0.68  inch)  thick.  According  to 
this,  the  sun's  heat  would  melt  about  158  feet  of  ice  annually  on  the 
earth's  equator,  or  124  feet  yearly  over  the  earth's  entire  surface  if 
the  heat  were  equally  distributed  in  all  latitudes. 
(See  note  at  the  end  of  the  chapter,  page  247.) 


FIG.  122*.  —  Bolograph  of  the  Infra-red  Spectrum,  Langley.    From  "  The  Sun," 
by  permission  of  D.  Appleton  &  Co. 

345.    Solar  Heat  Expressed  as  Energy. —  Since,  according  to  the 
known  value  of  the  "mechanical  equivalent  of  heat"  (Physics,  p.  199), 


SOLAK   KADIATION   AT   THE   SUN'S    SURFACE.  239 

a  horse-power  corresponds  to  about  10T7^  calories  per  minute,  it  fol- 
lows that  each  square  metre  of  surface  (neglecting  the  air-absorption) 
would  receive,  when  the  sun  is  overhead,  a  little  less  than  two  horse- 
power continuously.  Atmospheric  absorption  cuts  this  down  to 
about  one  and  one-fourth  horse-power,  of  which  about  one-eighth 
can  be  actually  utilized  by  properly  constructed  machinery,  as,  for 
instance,  the  solar  engines  of  Ericsson  and  Mouchot  (see  Langley's 
"New  Astronomy ").  In  Ericsson's  apparatus  the  reflector,  about 
11  feet  by  16  feet,  collected  heat  enough  to  work  a  three-horse- 
power engine  very  well.  Taking  the  earth's  surface  as  a  whole,  the 
energy  received  during  a  year  aggregates  about  a  hundred  mile-tons  for 
every  square  foot.  That  is  to  say,  the  heat  annually  received  on  each 
square  foot  of  the  earth's  surface,  if  employed  in  a  perfect  heat  engine, 
would  be  able  to  hoist  about  a  hundred  tons  to  the  height  of  a  mile. 

346.  Solar  Radiation  at  the  Sun's  Surface.  —  If,  now,  we  estimate 
the  amount  of  radiation  at  the  sun's  surface  itself,  we  come  to  results 
which  are  simply  amazing  and  beyond  comprehension.  It  is  neces- 
sary to  multiply  the  solar  constant  observed  at  the  earth  (which  is 
at  a  distance  of  93  000000  miles  from  the  sun)  by  the  square  of  the 
ratio  between  93  000000  and  433250,  the  radius  of  the  sun.  This 
square  is  about  46000  ;  in  other  words,  the  amount  of  heat  emitted 
in  a  minute  by  a  square  metre  of  the  sun's  surface  is  about  46000 
times  as  great  as  that  received  by  a  square  metre  at  the  earth. 
Carrying  out  the  calculations,  we  find  that  this  heat  radiation  at  the 
surface  of  the  sun  amounts  to  nearly  1  000000  calories  per  square 
metre  per  minute;  that  it  is  about  90000  horse-power  per  square 
metre  continuously  acting ;  that  if  the  sun  were  frozen  over  completely 
to  a  depth  of  forty-jive  feet,  the  heat  emitted  is  sufficient  to  melt  this 
whole  shell  in  one  minute  of  time;  that  if  an  ice  bridge  could  be 
formed  from  the  earth  to  the  sun  by  a  column  of  ice  two  and  one- 
half  miles  square  at  the  base  and  extending  across  the  whole 
93  000000  of  miles,  and  if  by  some  means  the  whole  of  the  solar 
radiation  could  be  concentrated  upon  this  column,  it  would  be  melted 
in  one  second  of  time,  and  in  between  seven  and  eight  seconds  more 
would  be  dissipated  in  vapor.  To  maintain  such  a  development  of 
heat  by  combustion  would  require  the  hourly  burning  of  a  layer  of  the 
best  anthracite  coal  from  nineteen  to  twenty-four  feet  thick  over  the 
sun's  entire  surface,  —  over  a  ton  for  every  square  foot,  —  at  least 
ten  times  as  much  as  the  consumption  of  the  most  powerful  blast 
furnace  in  existence.  At  that  rate  the  sun,  if  made  of  solid  coal, 
would  not  last  5000  years. 


240  THE    SUN. 

347.  Waste  of  Solar  Heat,  —  These  estimates  are  of  course  based  on 
the  assumption  that  the  sun  radiates  heat  equally  in  all  directions,  and  there 
is  no  assignable  reason  why  it  should  not  do  so.     On  this  assumption,  how- 
ever, so  far  as  we  can  see,  only  a  minute  fraction  of  the  whole  radiation  ever 
reaches  a  resting-place.     The  earth  receives  about  ^tnroVoui)  o  °^  ^e  wno^e' 
and  the  other  planets  of  the  solar  system,  with  the  comets  and  the  meteors, 
get  also  their  shares ;  all  of  them  together,  perhaps  ten  or  twenty  times  as 
much  as  the  earth.     Something  like  Too'oVunira  °^  tne  whole  seems  to  be 
utilized  within  the  limits  of  the  solar  system.     As  for  the  rest,  science  can- 
not yet  tell  what  becomes  of  it.     A  part,  of  course,  reaches  distant  stars  and 
other  objects  in  interstellar  space  ;  but  by  far  the  larger  portion  seems  to  be 
"wasted,"  according  to  our  human  ideas  of  waste. 

348.  Experiments  with  the  thermopile,  first  conducted  by  Henry 
at  Princeton  in  1845,  show  that  the  heat  from  the  edges  of  the  sun's 
disc,  like  the  light,  is  less  than  that  from  the  centre  —  according  to 
Langley's  measurements  about  half  as  much.     The  explanation  evi- 
dently lies  in  its  absorption  by  the  solar  atmosphere. 

349.  The  Sun's  Temperature.  —  While  we  can  measure  with  some 
accuracy  the  quantity  of  heat  sent  us  by  the  sun,  it  is  different  with 
its  temperature,  in  respect  to  which  we  can  only  say  that  it  must  be 

very  high — much  higher 
than  any  temperature 
attainable  by  known 
methods  on  the  surface 
of  the  earth. 

This   is   shown   by  a 

FIG.  123.  number  of  facts,  for  in- 

stance, by  the  great  abun- 
dance of  the  violet  and  ultra-violet  rays  in  the  sunlight. 

Again,  by  the  penetrating  power  of  sunlight ;  a  large  percentage 
of  the  heat  from  a  common  fire,  for  instance,  being  stopped  by  a 
plate  of  glass,  while  nearly  the  whole  of  the  solar  radiation  passes 
through. 

The  most  impressive  demonstration,  however,  follows  from  this 
fact  ;  viz.,  that  at  the  focus  of  a  powerful  burning-lens  all  known 
substances  melt  and  vaporize,  as  in  an  electric  arc.  Now  at  the 
focus  of  the  lens  the  limit  of  the  temperature  is  that  which  would 
be  produced  by  the  sun's  direct  radiation  at  a  point  where  the  sun's 
'angular  diameter  equals  that  of  the  burning-lens  itself  seen  from 
the  focus,  as  represented  in  Fig.  123.  An  object  at  F  would  theo- 
retically (that  is,  if  there  was  no  loss  of  heat  conducted  away  by 


EFFECTIVE    TEMPERATURE.  241 

surrounding  bodies  and  by  the  atmosphere)  reach  the  same  tempera- 
ture as  if  carried  to  a  point  where  the  sun's  angular  diameter  equals 
the  angle  LFL\  In  the  most  powerful  burning-lenses  yet  constructed 
a  body  at  the  focus  is  thus  virtually  carried  up  to  within  about 
240000  miles  of  the  sun's  surface,  where  its  apparent  diameter  would 
be  about  80°.  Here,  as  has  been  said,  the  most  refractory  substances 
are  immediately  subdued.  If  the  earth  were  to  approach  the  sun  as 
near  as  the  moon  is  to  us,  she  would  melt  and  be  vaporized. 

350,  Ericsson  in  1872  made  an  exceedingly  ingenious  and  interesting 
experiment  illustrating  the  intensity  of  the  solar  heat.     He  floated  a  calo- 
rimeter, containing  about  ten  pounds  of  water,  upon  the  surface  of  a  large 
mass  of  molten  iron  by  means  of  a  raft  of  fire-brick,  and  found  that  the 
radiation  of  the  metal  was  a  trifle  over  250  calories  per  minute  for  each 
square  foot  of  surface  ;  which  is  only  -^^  part  of  the  amount  emitted  by 
the  same  area  of  the  sun's  surface.     He  estimated  the  temperature  of  the 
metal  at  3000°  F.  or  1649°  C. 

351,  Effective  Temperature.  —  The  question  of  the  sun's  tempera- 
ture is  embarrassed  by  the  fact  that  it  has  no  one  temperature ;   the 
temperature  at  different  parts  of  the  solar  photosphere  and  chromo- 
sphere must  be  very  different.     We  evade  this  difficulty  to  some 
extent  by  substituting  for  the  actual  temperature,  as  the  object  of 
inquiry,  what  has  been  called  the  sun's  "effective  temperature";  that 
is,  the  temperature  which  a  sheet  of  lampblack  must  have  in  order 
to  radiate  the    amount  of   heat    actually  thrown  off   by  the    sun. 
(Physicists  have  taken  the  radiating  power  of  lampblack  as  unity.) 
If  we  could  depend  upon  the  laws  J  deduced  from  laboratory  experi- 
ments, by  which  it  has  been  sought  to  connect  the  temperature  of 
the  body  with  its  rate  of  radiation,  the  matter  would  then  be  com- 
paratively simple :    from  the  known  radiated  quantity  of  heat  (in 
calories)  we  could  compute  the   effective  temperature   in   degrees. 
But  at  present  it  is  only  by  a  very  unsatisfactory  process  of  extra- 
polation that  we  can  reach  conclusions.     The  sun's  temperature  is 
so  much  higher  than  any  which  we  can  manage  in  our  laboratories, 
that  there  is  not  yet  much  certainty  to  be  obtained  in  the  matter. 

1  A  number  of  such  laws  have  been  formulated ;  for  instance,  the  well-known 
law  of  Dulong  and  Petit.  Pouillet  and  Vicaire,  using  this  formula,  have  deduced 
values  for  the  sun's  effective  temperature  ranging  from  1500°  and  2500°  C. 
Ericsson  and  Secchi,  using  Newton's  law  of  radiation  (which,  however,  is  cer- 
tainly inapplicable  under  the  circumstances),  put  the  figure  among  the  millions. 
Wilson  and  Gray's  results  agree  nearly  with  Stefan's  "fourth  power"  law  ;  viz., 
t  =  kt*,  t  being  the  "  absolute  "  temperature,  and  k  a  constant,  depending  on 
the  substance  which  radiates. 


242  THE   SUN. 

Wilson  and  Gray,  the  most  recent  and  reliable  investigators,  from 
their  work  in  1894-95,  get  8000°  C.  or  14440°  F.  for  the  effective 
temperature.  Almost  certainly  it  lies  somewhere  between  10000° 
and  20000°  F. 

352.  Constancy  of  the  Sun's  Heat.  —  It  is  an  interesting  and  thus 
far  unsolved  problem,  whether  the  total  amount  of  the  sun's  radia- 
tion varies  perceptibly  at  different  times.     It  is  only  certain  that 
the  variations,  if  real,  are  too  small  to  be  detected  by  our  present 
means  of  observation.     Possibly,  at  some  time  in  the  future,  ob- 
servations on  a  mountain  summit  above  the   main  body  of   our 
atmosphere  may  decide  the  question. 

It  is  not  unlikely  that  changes  in  the  earth's  climate  such  as 
have  given  rise  to  glacial  and  carboniferous  periods  may  ultimately 
be  traced  to  the  condition  of  the  sun  itself,  especially  to  changes  in 
the  thickness  of  the  absorbing  atmosphere,  which,  as  Langley  has 
pointed  out,  must  have  a  great  influence  in  the  matter.  Since  the 
Christian  era,  however,  it  is  certain  that  the  amount  of  heat  annually 
received  from  the  sun  has  remained  practically  unchanged.  This  is 
inferred  from  the  distribution  of  plants  and  animals,  which  is  still 
substantially  the  same  as  in  the  days  of  Pliny. 

353.  Maintenance  of  the  Solar  Heat.  —  The  question  at  once 
arises,  if  the  sun  is  sending  off  such  an  enormous  quantity  of  heat 
annually,  how  is  it  that  it  does  not  grow  cold  ? 

(a)  The  sun's  heat  cannot  be  kept  up  by  combustion.     As  has 
been  said  before,  it  would  have  burned  out  long  ago,  even  if  made 
of  solid  coal  burning  in  oxygen. 

(b)  Nor  can  it  be  simply  a  heated  body  cooling  down.     Huge  as  it 
is,  an  easy  calculation  shows  that  its  temperature  must  have  fallen 
greatly  within  the  last  2000  years  by  such  a  loss  of  heat,  even  if  it 
had  a  specific  heat  higher  than  that  of  any  known  substance. 

As  matters  stand  at  present,  the  available  theories  seem  to  be 
reduced  to  two,  —  that  of  Mayer,  which  ascribes  the  solar  heat  to 
the  energy  of  meteoric  matter  falling  on  the  sun ;  and  that  of  Helm- 
holtz,  who  finds  the  cause  in  a  slow  contraction  of  the  sun's  diameter. 

354.  Meteoric  Theory  of  Sun's  Heat.  —  The  first  is  based  on  the 
fact  that  when  a  moving  body  is  stopped,  its  mass-energy  becomes 
molecular  energy,  and  appears  mainly  as  heat.     The  amount  of  heat 
developed  in  such  a  case  is  given  by  the  formula 


8339 


OBJECTIONS   TO   METEORIC    THEORY    OF   SUN'S   HEAT.     243 

in  which  Q  is  the  number  of  calories  of  heat  produced,  M  the  mass 
of  the  moving  body  in  kilograms,  and  V  its  velocity  in  metres  per 
second;  the  denominator  is  the  "mechanical  equivalent  of  heat" 
(Physics,  p.  98)  multiplied  by  2g  expressed  in  metres  ;  i.e., 
425  X  2  X  9.81. 

Now,  the  velocity  of  a  body  coming  from  any  considerable  dis- 
tance and  falling  into  the  sun  can  be  shown  to  be  about  380  miles 
per  second,  or  more  than  610  kilometres.  A  body  weighing  one 
kilogram  would  therefore,  on  striking  the  sun  with  this  velocity, 
produce  about  45  000000  calories  of  heat, 

r(610000)2"| 
L     8339     J 

This  is  6000  times  more  than  could  be  produced  by  burning  it,  even  if 
it  were  coal  or  solidified  hydrogen  burning  in  pure  oxygen. 

Now,  as  meteoric  matter  is  continually  falling  upon  the  earth,  it 
must  be  also  falling  upon  the  sun,  and  in  vastly  greater  quantities, 
and  an  easy  calculation  shows  that  a  quantity  of  meteoric  matter 
equal  to  TL  of  the  earth's  mass  striking  the  sun's  surface  annually 
with  the  velocity  of  600  kilometres  per  second  would  account  for  its 
whole  radiation. 

355.  Objections  to  Meteoric  Theory  of  Sun's  Heat.  —  There  can  be 
no  question  that  a  certain  fraction  of  the  sun's  heat  is  obtained  in 
this  way,  but  it  is  very  improbable  that  this  fraction  is  a  large  one ; 
indeed,  it  is  hardly  possible  that  it  can  be  as  much  as  one  per  cent 
of  the  whole. 

(1)  The  annual  fall  on  the  sun's  surface  of  such  a  quantity  of  meteoric 
matter  implies  the  presence  near  the  sun  of  a  vastly  greater  mass  ;  for,  as 
we  shall  see  hereafter,  only  a  few  of  the  meteors  that  approach  the  sun  from 
outer  space  would  strike  the  surface  ;  most  of  them  would  act  like  the  comets 
and  swing  around  it  without  touching.     Now,  if  there  were  any  considerable 
quantity  of  such  matter  near  the  sun,  there  would  result  disturbances  in  the 
motions  of  the  planets  Mercury  and  Venus,  such  as  observation  does  not  reveal 

(2)  Professor  Peirce  has  shown  further  that  if  the  heat  of  the  sun  were 
produced  in  this  way,  the  earth  ought  to  receive   from   the  meteors  that 
strike  her  surface  about  half  as  much  heat  as  she  gets  from  the  sun.     Now 
the  quantity  of  meteoric  matter  which  would  have  to  fall  upon  the  earth  to 
furnish  us  daily  half  as  much  heat   as  we  receive  from  the  sun  would 
amount  to  nearly  fifty  tons  for  each  square  mile.     It  is  not  likely  that  we 
actually  get  lirffiTnnnj.  of  that  amount.     It  is  difficult  to  determine  the 
amount  of  heat  which  the  earth  actually  does  receive  from  meteors,  but  all 
observations  indicate  that  the  quantity  is  extremely  small.     The  writer  has 


244  THE   SUN. 

estimated  it,  from  the  best  data  attainable,  as  less  in  a  year  than  we  get 
from  the  sun  in  a  second. 

356,  Helmholtz's  Theory  of  Solar  Contraction.  —  We  seem  to  be 
shut  up  to  the  theory  of  Helmholtz,  now  almost  universally  accepted ; 
namely,  that  the  heat  necessary  to  maintain  the  sun's  radiation  is 
principally  supplied  by  the  slow  contraction  of  its  bulk,  aided,  how- 
ever, by  the  accompanying  liquefaction  and  solidification  of  portions 
of  its  gaseous  mass.     When  a  body  falls  through  a  certain  distance, 
gradually,  against  resistance,  and  then  comes  to  rest,  the  same  total 
amount  of  heat  is  produced  as  if  it  had  fallen  freely,  and  been 
stopped  instantly.     If,  then,  the  sun  does  contract,  heat  is  necessarily 
produced  by  the  process,  and  that  in  enormous  quantity,  since  the 
attracting  force  at  the  solar  surface  is  more  than  twenty-seven  times 
as  great  as  terrestrial  gravity,  and  the  contracting  mass  is  immense. 
In  this  process  of  contraction  each  particle  at  the  surface  moves 
inward  by  an  amount  equal  to  the  diminution  of  the  sun's  radius  : 
a  particle  below  the  surface  moves  less  and  under  a  diminished 
gravitating  force ;  but  every  particle  in  the  whole  mass,  excepting 
only  that  at  the  exact  centre  of  the  globe,  contributes  something  to 
the  evolution  of  heat.     In  order  to  calculate  the  precise  amount  of 
heat  evolved  by  a  given  shrinkage,  it  would  be  necessary  to  know 
the  law  of  increase  of  the  sun's  density  from  the  surface  to  the 
centre  ;    but  Helmholtz  has  shown  that  under  the  most  unfavorable 
conditions  a  contraction  of  the  sun's  diameter  of  about  60  metres  or 
200  feet  a  year  (100  feet  in  the  sun's  radius)  would  account  for  the 
whole  annual  output  of  heat.     This  contraction  is  so  slow  that  it 
would  be  quite  imperceptible  to  observation.     It  would  require  very 
nearly  10000  years  to  reduce  the  sun's  diameter  by  a  single  second  of 
arc ;  and  nothing  much  less  would  be  certainly  detectable  by  our 
measurements.     If  the  contraction  is  more  rapid  than  this,  the  mean 
temperature  of  the  sun  must  be  actually  rising,  notwithstanding  the 
amount  of  heat  it  is  losing.     Long  observation  alone  can  determine 
whether  this  is  really  the  case  or  not. 

357.  Lane's  Law.  —  It  is  a  remarkable  fact,  first  demonstrated  by  Lane 
of  Washington,  in  1870,  that  a  gaseous  sphere,  losing  heat  by  radiation  and 
contracting  under  its  own  gravity,  must  rise  in  temperature  and  actually  grow 
hotter,  until  it  ceases  to  be  a  "  perfect  gas,"  either  by  beginning  to  liquefy, 
or  by  reaching  a  density  at  which  the  laws  of  perfect  gases  no  longer  hold. 
The  kinetic  energy  developed  by  the  shrinkage  of  a  gaseous  mass  is  more 
than  sufficient  to  replace  the  loss  of  heat  which  caused  the  shrinkage.     In 
the  case  of  a  solid  or  liquid  mass  this  is  not  so.     The  shrinkage  of  such  a 
mass  contracting  under  its  own  gravity  on  account  of  the  loss  of  heat  is 


FUTURE   DURATION    OF    THE    SUN.  245 

never  sufficient  to  make  good  the  loss ;  but  the  temperature  falls  and  the 
body  cools.  At  present  it  appears  that  in  the  sun  the  relative  proportions 
of  true  gases  and  liquids  are  such  as  to  keep  the  temperature  nearly  station- 
ary, the  liquid  portions  of  the  sun  being  of  course  the  little  drops  which  are 
supposed  to  constitute  the  clouds  of  the  photosphere. 

358.  Future  Duration  of  the  Sun. —  If  this  shrinkage  of  the  sun's 
diameter  has  been  the  only  source  of  solar  heat,  it  will  follow  in 
time  that  the  sun's  heat  must  come  to  an  end,  and,  looking  back- 
wards, we  see  that  there  must  have  been  a  beginning. 

We  have  not  sufficient  data  to  enable  us  to  calculate  the  future 
duration  of  the  sun  with  exactness,  though  an  approximate  estimate 
can  be  made.  According  to  ISTewcomb,  if  the  sun  maintains  its 
present  radiation,  it  will  have  shrunk  to  half  its  present  diameter 
in  about  5  000000  years  at  the  longest.  Since,  when  reduced  to  this 
size,  it  must  be  about  eight  times  as  dense  as  now,  it  can  hardly  then 
continue  to  be  mainly  gaseous,  and  its  temperature  must  begin  to 
fall.  Newcomb's  conclusion,  therefore,  is  that  it  is  not  likely  that 
the  sun  can  continue  to  give  sufficient  heat  to  support  such  life  on 
the  earth  as  we  are  now  acquainted  with,  for  10  000000  years  from 
the  present  time. 

359.  Age  of  the  Sun.  —  As  to  the  past  of  the  solar  history  on  this 
hypothesis,  we  can  be  a  little  more  definite.     It  is  only  necessary  to 
know  the  present  amount  of  radiation,  and  the  mass  of  the  sun,  to 
compute  how  long  the  solar  fire  can  have  been  maintained  at  its 
present  intensity  by  the  processes  of  condensation.     No  conclusion 
of  geometry  is  more  certain  than  this,  —  that  the  contraction  of  the 
sun  to  its  present  size,  from  a  diameter  even  many  times  greater 
than  Neptune's  orbit,  would  have  furnished  about  18  000000  times 
as  much  heat  as  the  sun  now  supplies  in  a  year,  and  therefore  that 
the  sun  cannot  have  been  emitting  heat  at  the  present  rate  for  more 
than  18  000000  years,  if  its  heat  has  really  been  generated  in  this 
manner. 

But  this  conclusion  rests  upon  the  assumption  that  the  sun  has  derived 
its  heat  solely  in  this  way,  and  the  recent  discoveries  with  respect  to  radium 
and  radio-activity  strongly  suggest  other  causes  which  may  have  added  large 
contributions  and  may  still  be  operative  in  maintaining  the  solar  radiation. 

360.  Constitution  of  the  Sun.  —  (a)  As  to  the  nature  of  the  main 
body  or  nucleus  of  the  sun,  we  cannot  be  said  to  have  certain  knowl- 
edge.    It  is  probably  gaseous,  this  being  indicated  by  its  low  mean 
density  and  its  high  temperature  —  enormously  high  even  at  the 


246  THE   SUN. 

surface,  where  it  is  coolest.  At  the  same  time  the  gaseous  matter 
at  the  nucleus  must  be  in  a  very  different  state  from  gases  as  we 
commonly  know  them  in  our  laboratories,  on  account  of  the  intense 
heat  and  the  extreme  condensation  by  the  enormous  force  of  solar 
gravity.  The  central  mass,  while  still  strictly  gaseous,  because  ob- 
serving the  three  physical  laws  of  Boyle,  Dalton,  and  Gay  Lussac, 
which  characterize  gases,  would  be  denser  than  water,  and  viscous ; 
probably  something  like  tar  or  pitch  in  consistency.1 

While  this  doctrine  of  the  gaseous  constitution  of  the  sun  is  gener- 
ally assented  to,  there  are  still  some  who  are  disposed  to  consider 
the  great  mass  of  the  sun  as  liquid. 

361.  (#)    The  photosphere  is  probably  a  shell   of   incandescent 
clouds,  formed  by  the  condensation  of  the  vapors  which  are  exposed 
to  the  cold  of  space. 

The  minute  particles  of  which  the  photosphere  is  composed  being  liquid, 
or  possibly  some  of  them  solid,  have  a  radiating  power  enormously  greater 
than  that  of  the  gases  in  which  they  float,  though  the  temperature  is  practi- 
cally the  same.  As  a  source  of  light  and  heat  the  photosphere  acts  in  the 
same  way  as  the  "mantle"  of  a  Welsbach  burner. 

362.  (c)    The  photospheric  clouds  float  in  an  atmosphere  contain- 
ing, still  uncondensed,  a  considerable  quantity  of  the  same  vapors 
out  of  which  they  themselves  have  been  formed,  just  as  in  our  own 
atmosphere  the  air  around  a  cloud  is  still  saturated  with  water  vapor. 
This  vapor-laden  atmosphere,  probably  comparatively  shallow,  con- 
stitutes the  reversing  layer,  and  by  its  selective  absorption  produces 
the  dark  lines  of  the  solar  spectrum,  while  by  its  general  absorption 
it  probably  produces  the  darkening  at  the  limb  of  the  sun. 

But  it  will  be  remembered  that  Mr.  Lockyer  and  others  are  disposed  to 
question  the  existence  of  any  such  shallow  absorbing  stratum,  considering 
that  the  absorption  takes  place  in  all  regions  of  the  solar  atmosphere  even 
to  a  great  elevation. 


1  The  law  of  Dalton  (Physics,  p.  218)  is,  that  any  number  of  different  gases 
and  vapors  tend  to  distribute  themselves  throughout  the  space  which  they  occupy  in 
common,  each  as  if  the  others  were  absent.  The  law  of  Boyle  or  Mariotte  (Physics, 
p.  118)  is,  that  at  any  given  temperature  the  volume  of  any  given  amount  of  gas 
varies  inversely  with  the  pressure;  i.e.,  pv  =  p'v'.  The  law  of  Gay  Lussac 
(Physics,  p.  222)  is,  that  a  gas  under  constant  pressure  expands  in  volume  uni- 
formly under  uniform  increment  of  temperature,  so  that  Vt  =  VQ  (1  +  at).  This 
is  not  true  of  vapors  in  presence  of  the  liquids  from  which  they  have  been  evap- 
orated ;  for  instance,  of  steam  in  a  boiler. 


CONSTITUTION   OF   THE   SUN.  247 

363.  (d)    The  chromosphere  and  prominences  are  composed  of  the 
permanent  gases,  mainly  hydrogen  and  helium,  which  are  mingled 
with  the  vapors  of  the  reversing  stratum  in  the  region  near  the 
photosphere,  but  usually  rise  to  far  greater  elevations  than  do  the 
vapors.     The  appearances  are  for  the  most  part  as  if  the  chromo- 
sphere was  formed  of  jets  of  heated  hydrogen  ascending  through 
the  interspaces  between  the  photospheric  clouds,  like  flames  playing 
over  a  coal  fire. 

364.  (e)  The  corona  also  rests  on  the  photosphere,  and  the  pecu- 
liar green  line  of  its  spectrum  (Art.  329)  is  brightest  just  at  the 
surface  of  the  photosphere,  in  the  reversing  stratum  and  in  the 
chromosphere  itself ;    but  the  corona  extends  to  a  far  greater  eleva- 
tion than  even  the  prominences  ever  reach,  and  seems  to  be  not 
wholly  gaseous,  but  to  contain,  besides  the  hydrogen  and  the  mys- 
terious "coronium,"  dust  and  fog  of  some  sort,  perhaps  meteoric. 
Many  of  its  phenomena  are  as  yet  unexplained,  and  since  it  can 
only  be  observed  during  the  brief  moments  of  total  solar  eclipses, 
progress  in  its  study  is  necessarily  slow. 

364*.  (Note  to  Art.  344.)  Taking  the  solar  constant  at  21  cal.  per 
square  metre  per  minute,  the  amount  of  heat  falling  upon  a  square  metre 
in  an  hour  would  raise  1260  kilograms  (or  cubic  decimetres)  of  water  1°  C. 
in  temperature.  Since  the  "  heat  of  fusion  "  of  ice  is  79.25,  this  would  melt 
^£|£,  or  15. 9^  kgms.  of  ice;  and  the  specific  gravity  of  ice  being  0.92,  this 
would  correspond  to  £:5^f,  or  17.3,  cubic  decimetres,  which  spread  over  a 
square  metre  would  make  a  thickness  of  17.3mm  or  0.68in. 

The  total  heat  received  by  the  earth  is  that  intercepted  by  its  diametrical 
section,  or  the  area  of  one  of  its  great  circles. 

The  thickness  of  the  sheet  of  ice  melted  annually  upon  this  circular  plane 
would  be  17.3mm  x  24  x  365 $  =  151.5  metres,  or  497  feet.  On  a  narrow 
equatorial  belt  the  thickness  melted  would  be  li^sm  =  48.2  metres,  or  158 
feet,  since  such  a  belt  intercepts  the  rays  that  otherwise  would  fall  upon  a 
diametrical  strip  of  the  same  width  on  the  circular  plane.  If  the  sun's  heat 
were  uniformly  distributed  over  the  whole  surface  of  the  earth,  the  area  of 
which  equals  four  great  circles  (47rr2),  it  could  melt  an  ice  sheet  having  a 
thickness  of  J-5^m,  or  37. 9m  (124.2  feet).  It  must  be  remembered,  how- 
ever, that  the  value  of  the  solar  constant  is  likely  to  be  in  error,  perhaps  as 
much  as  ten  per  cent ;  and  all  the  numbers  above  given  are  affected  by  the 
same  uncertainty. 

It  is  true  that  at  the  sea-level  the  solar  constant  is  much  diminished  by 
atmospheric  absorption;  and  probably  does  not  equal  eighteen  calories  per 
minute  directly  received  from  the  sun's  rays.  But  a  large  part  of  the  solar 
heat  absorbed  by  the  atmosphere  reaches  the  earth's  surface  indirectly  through 
the  warming  of  the  atmosphere,  so  that  it  must  not  be  considered  as  lost  to 
the  earth  because  not  directly  measurable  by  the  actinometer. 


248  ECLIPSES. 


CHAPTER   XL 

ECLIPSES :  FORM  AND  DIMENSIONS  OF  SHADOWS.  —  LUNAR 
ECLIPSES.  —  SOLAR  ECLIPSES.  —  TOTAL,  ANNULAR,  AND  PAR- 
TIAL. —  ECLIPTIC  LIMITS  AND  NUMBER  OF  ECLIPSES  IN  A 
YEAR.  —  THE  SAROS.  —  OCCULTATIONS. 


365.  THE  word  eclipse  (Greek  e/cAet^t?)  is  strictly  a  medical  term, 
meaning  a  faint  or  swoon.     Astronomically  it  is  applied  to  the  dark- 
ening of   a  heavenly  body,  especially  of  the  sun  or  moon,  though 
some  of  the  satellites  of   other   planets  besides  the  earth  are  also 
"  eclipsed"  from  time  to  time.     An  eclipse  of  the  moon  is  caused 
by  its  passage  through  the  shadow  of  the  earth  ;    an  eclipse  of  the 
sun^  by  the  interposition  of  the  moon  between  the  sun  and  the  ob- 
server, or,  what   comes  to  the  same   thing,  by  the  passage  of   the 
moon's  shadow  over  the  observer. 

366.  Shadows.  —  If  interplanetary  space  were  slightly  dusty,  we 
should    see,   accompanying   the    earth   and    moon    and    each   of  the 
planets,   a  long   black  shadow  projecting    behind    it    and   travelling 
with  it.     Geometrically  speaking,  this  shadow  of  a  body,  the  earth 
for   instance,  is  a  solid  —  not  a  surface.     It  is  the  space  from  which 
sunlight    is    excluded.      If    we  regard  the   sun  and   other   heavenly 
bodies  as  truly  spherical,  these  shadows   are  cones  with  their  axes 
in  the  line    joining  the  centres  of   the  sun  and  the  shadow-casting 
body,  the  point  being  always  directed  away  from  the  sun,  because 
the  sun  is  always  the  larger  of  the  two. 

367.  Dimensions   of  the   Earth's   Shadow. — The   length  of   the 
shadow  is  easily  found.     In    Fig.    124   we   have    from   the    similar 
triangles  OED  and  ECa,    OD  :  Ea  : :  OE :  EC  or  1.     OD  is  the  dif- 
ference   between    the    radii   of    the    sun    and   the    earth,  =  7?  — r. 
Ea  =  r,  and  OE  is  the  distance  of  the  earth  from  the  sun  =  A. 

Hence 

(The  fractional  factor  is  constant,  since  the  radii  of  the  sun  and 


PENUMBRA.  249 

earth  are  fixed  quantities.  Substituting  the  values  of  the  radii,  we 
find  it  to  be  yoVir)  This  gives  857200  miles  for  the  length  of  the 
earth's  shadow  when  A  has  its  mean  value  of  93  000000  miles,  regard- 


FIG.  124.  —  Dimensions  of  the  Earth's  Shadow. 

ing  the  earth  as  a  perfect  sphere  and  taking  its  mean  radius.  This 
length  varies  about  14000  miles  on  each  side  of  the  mean  as  the 
earth  changes  its  distance  from  the  sun. 

The  semi-angle  of  the  cone  (the  angle  ECb,  or  ECB  in  the  figure.)  is 
found  as  follows:  Since  OEB  is  exterior  to  the  triangle  EEC, 

OEB  =  EEC  4-  BCE, 
or 

BCE  =  OEB  -  EBC. 

Now,  OEB  is  the  sun's  apparent  semi-diameter  as  seen  from  the  earth,  and 
EBC  is  the  earth's  semi-diameter  as  seen  from  the  sun,  which  is  the  same 
thing  as  the  sun's  horizontal  parallax  (Art.  83). 

Putting  S  for  the  sun's  semi-diameter,  and  p  for  its  parallax,  we 
have  — 

Semi-angle  at  C=  S— p.1 

From  the  cone  a  Cb  all  sunlight  is  excluded,  or  would  be  were  it 
not  for  the  fact  that  the  atmosphere  of  the  earth  by  its  refraction 
bends  some  of  the  rays  into  this  shadow.  The  effect  is  to  make  the 
shadow  a  little  larger  in  diameter,  but  less  perfectly  dark. 

368,  Penumbra.  —  If  we  draw  the  lines  Ba  and  Ab,  crossing  at 
C1  between  the  earth  and  the  sun,  they  will  bound  the  penumbra. 
Within  this  space  a  part,  but  not  the  whole,  of  the  sunlight  is  cut 
off :  an  observer  outside  of  the  shadow,  but  within  this  cone-frustum, 


1  Also,  I  =  — — — -r,  an  expression  sometimes  more  convenient  than  the  one 

sin  (o      p) 

given  above. 


250  ECLIPSES. 


which  tapers  towards  the  sun,  would  see  the  earth  as  a  black  body 
encroaching  on  the  sun's  disc.  The  semi-angle  of  the  penumbra  EC'a 
is  easily  shown  to  be  S 


369.  Although  geometrically  the  boundaries  of  the  shadow  and 
penumbra  are  perfectly  definite,  they  are  not  so  optically.     If  a  screen 
were  placed  at  M  (Fig.  124)  perpendicular  to  the  axis  of  the  shadow, 
no  sharply  defined  lines  would  mark  the  boundaries  of  either  shadow 
or  penumbra  ;  near  the  edge  of  the  shadow,  the  penumbra  would 
be  very  nearly  as  dark  as  the  shadow  itself,  only  a  mere  speck  of  the 
sun  being  visible  there  ;  and  at  the  outer  limit  of  the  penumbra  the 
shading  would  be  still  more  gradual. 

370.  Eclipses  of  the  Moon.  —  The  axis  of  the  earth's  shadow  is 
always  directed  to  a  point  exactly  opposite  the  sun.     If,  then,  at  the 
time  of  full  moon,  the  moon  happens  to  be  near  the  ecliptic  (that  is, 
not  far  from  one  of  the  nodes  of  her  orbit)  ,  she  will  pass  into  the 
shadow  and  be  eclipsed.     Since,  however,  the  moon's  orbit  is  inclined 
about  five  and  one-fourth  degrees  to  the  plane  of  the  ecliptic,  this 
does  not  happen  very  often  (seldom  more  than  twice  a  year)  .     Ordi- 
narily the  moon  passes  north  or  south  of  the  shadow  without  touch- 
ing it. 

Lunar  eclipses  are  of  two  kinds,  —  partial  and  total  :  total  when 
the  moon  passes  into  the  shadow  completely  ;  partial  when  she  goes 
so  far  to  the  north  or  south  of  the  centre  of  the  shadow  that  only  a 
portion  of  her  disc  is  obscured. 

We  may  also  have  a  "  penumbral  eclipse  "  when  she  passes  merely  through 
the  penumbra  without  touching  the  shadow.  In  this  case,  however,  the  loss 
of  light  is  so  gradual  and  so  slight,  unless  she  almost  grazes  the  shadow, 
that  an  observer  would  notice  nothing  unusual. 

371.  Size  of  the  Earth's  Shadow  at  the  Point  where  the  Moon 
crosses  it.  —  Since  EC  in  Fig.  125  is  857,000  miles,  and  the  distance 
of  the  moon  from  the  earth  is  on  the  average  about  239,000  miles, 
CM  must  be  618,000  miles,  and  MN,  the  semi-diameter  of  the  shadow 
at  this  point,  will  be  f^f  of  the  earth's  radius.    This  gives  MN=  2854 
miles,  and  makes  the  whole  diameter  of  the  shadow  a  little  over  5700 
miles,  about  two  and  two-thirds  times  the  diameter  of  the  moon.    But 
this  quantity  varies  considerably.     The  shadow  is  sometimes  more 
than  three  times  as  large  as  the  moon,  sometimes  hardly  more  than 
twice  its  size. 


DURATION  OF   LUNAR   ECLIPSE.  251 

372.  We  may  reach  the  same  result  in  another  way.  Considering  the 
triangle  ECN,  Fig.  125,  we  have  the  angular  semi-diameter  of  the  cross- 
section  of  the  shadow  where  the  moon  passes  through  it,  as  seen  from  the 
earth,  represented  by  MEN. 

But  ENa  =  MEN  +  ECN-, 

whence  MEN  =  ENa  —  E  CN. 

Now  ENa  is  the  semi-diameter  of  the  earth  as  seen  from  the  moon  ;  that 
is,  it  is  the  moon's  horizontal  parallax,  for  which  write  P.  Hence,  substituting 
for  ECN  its  value  S—p,  we  get 


=  P+p-S. 

MEN  is  called  "  the  radius  of  the  shadow."     The  mean  value  of  P  is  57'  2"  ; 
of  jo,  8".8  ;  and  of  S,  16'  2",  which  makes  the  mean  value  of  MEN=  41'  9". 


L.I.Jill 

FIG.  125.  —  Diameter  of  Earth's  Shadow  where  the  Moon  crosses  it. 

The  mean  value  of  the  moon's  apparent  semi-diameter  is  15'  40",  the  ratio 
between  the  semi-diameter  of  the  moon  and  the  radius  of  the  shadow  being 
about  2f,  as  before. 

In  computing  a  lunar  eclipse,  this  angular  value  for  the  "  radius  of  the 
shadow,"  as  it  is  called,  is  more  convenient  than  its  value  in  miles.  It  is 
customary  to  increase  it  by  about  ^  part  in  order  to  allow  for  the  effect  of 
the  earth's  atmosphere,  the  value  ordinarily  used  being  f£  (P  +  jo  — *S). 
Some  computers,  however,  use  |^,  and  others  ||.  On  account  of  the  indis- 
tinctness of  the  edge  of  the  shadow  it  is  not  easy  to  determine  what  precise 
value  ought  to  be  employed,  nor  is  it  important. 

373.  Duration  of  a  Lunar  Eclipse.  — When  central,  a  total  eclipse 
of  the  moon  may,  all  things  favoring,  continue  total  for  about  two 
hours,  the  interval  from  the  first  contact  to  the  last  being  about  two 
hours  more.  This  depends  upon  the  fact  that  the  moon's  hourly 
motion  is  nearly  equal  to  its  own  diameter.  The  whole  interval  from 
first  contact  to  last  is  the  time  occupied  by  the  moon  in  moving  from 


252 


ECLIPSES. 


a  to  d  (Fig.  126).  The  totality  lasts  while  it  moves  from  b  to  c. 
The  duration  of  a  non-central  eclipse  varies,  of  course,  according  to 
the  part  of  the  shadow  through  which  the  moon  passes. 


FIG.  126.  —  Duration  of  a  Lunar  Eclipse. 

374.  Lunar  Ecliptic  Limit.  —  The  lunar  ecliptic  limit  is  the  great- 
est distance  from  the  node  of  the  moon's  orbit  at  which  the  sun 
can  be  consistently  with  having  an  eclipse.    This  limit  depends  upon 
the  inclination  of  the  moon's  orbit,  which  varies  a  little,  and  also  upon 
the  radius  of  the  shadow  at  the  time  of  the  eclipse  and  the  moon's 
apparent  semi-diameter,  which  quantities  are  still  more  variable. 
Hence  we  recognize  two  limits,  the  major  and  minor.     If  the  dis- 
tance of  the  sun  from  the  node  at  the  time  of  full  moon  exceeds  the 
major  limit,  an  eclipse  is  impossible  ;  if  it  is  less  than  the  minor,  an 
eclipse  is  inevitable.  The  major  limit  is  found  to  be  12°  15';  the  minor, 
9°30r.     Since  the  sun  passes  over  an  arc  of  12°  15'  in  less  than 
thirteen  days,  it  follows  that  an  eclipse  of  the  moon  cannot  possibly 
take  place  more  than  thirteen  days  before  or  after  the  time  when  the 
sun  crosses  the  node. 

375.  In  Fig.  127  let  NE  and  NM  be,  respectively,  portions  of  the 
ecliptic  and  of  the  path  of  the  moon,  as  seen  projected  upon  the  celestial 


FIG.  127.  —  Lunar  Ecliptic  Limit. 

sphere.     E  is  the  centre  of  the  earth's  shadow.     The  sun,  of  course,  is  at 
the  point  of  the  celestial  sphere  directly  opposite,  and  its  distance  from  the 


PHENOMENA    OF    A    TOTAL   LUNAK    ECLIPSE.  253 

opposite  node  is  equal  to  EN.  M  is  the  centre  of  the  moon.  Call  the 
semi-diameter  of  the  moon  S' ;  then  EM  (the  greatest  possible  distance 
between  E  and  M  which  permits  an  eclipse)  equals  the  sum  of  the  semi- 
diameters  of  the  moon  and  shadow,  or  Sf  +  (P  +  p  —  S),  and  the  corre- 
sponding ecliptic  limit  EN  is  found  by  solving  the  spherical  triangle  MNE, 
having  given  ME  and  the  angle  at  .ZV,  which  is  about  5i°.  We  must  also 
know  one  other  angle,  and  with  sufficient  approximation  for  such  purposes 
we  may  regard  the  angle  at  M  as  a  right  angle.  The  limit  is  always  very 
nearly  eleven  times  EM,  because  the  inclination  of  the  moon's  orbit  is 
nearly  ^  of  a  "radian." 

376.  Phenomena  of  a  Total  Lunar  Eclipse.  —  Half  an  hour  or  so 
before  the  moon  reaches  the  shadow  its  eastern  limb  begins  to  be 
sensibly  darkened,  and  the  edge  of  the  shadow  itself,  when  it  is  first 
reached,  looks  nearly  black  by  contrast  with  the  bright  parts  of  the 
moon's  surface.  To  the  naked  eye  the  outline  of  the  shadow  appears 
reasonably  sharp ;  but  with  even  a  small  telescope  it  is  found  to  be 
indefinite  and  hazy,  and  with  a  large  instrument  and  high  magnify- 
ing power  it  becomes  entirely  indistinguishable.  It  is  impossible  to 
determine  the  exact  moment  when  the  edge  of  the  shadow  reaches 
any  particular  point  on  the  moon  within  half  a  minute  or  so. 


FIG.  128.  —  Light  Bent  into  Earth's  Shadow  by  Kefraction. 

After  the  moon  has  wholly  entered  the  shadow  her  disc  is  usually 
still  distinctly  visible,  illuminated  with  a  dull,  copper-colored  light, 
which  is  sunlight,  deflected  around  the  earth  into  the  shadow  by  the 
refraction  of  our  own  atmosphere,  or  rather  by  that  portion  of  our 
atmosphere  which  lies  within  ten  or  fifteen  miles  of  the  earth's  sur- 
face.    Since  the  ordinary  horizontal  refraction  is  34',  it  follows  that 
light  which  just  grazes  the  earth's  surface  will  be  bent  inwards  by 
twice  that  amount,  or  1°  S'.     Now,  the  maximum  parallax  *  of  the 
moon  is  only  1°  2'.     In  an  extreme  case,  therefore,  even  when  the 
moon  is  exactly  central  in  the  largest  possible  shadow,  it  receives 
some  sunlight  coming  around  the  edge  of  the  earth,  as  shown  by 
Fig.  128.     To  an  observer  stationed  on  the  moon,  the  disc  of  the 
earth  would  appear  to  be  surrounded  by  a  narrow  ring  of  brilliant 
sunshine,  colored  with  sunset  hues  by  the  same  vapors  which  tinge 

1  This  is  the  semi-diameter  of  the  earth  as  seen  from  the  moon. 


254  ECLIPSES. 

terrestrial  sunsets,  but  acting  with  double  power  because  the  light  has 
traversed  a  double  thickness  of  our  air.  If  the  weather  happens  to 
be  clear  at  this  portion  of  the  earth  (upon  its  rim  as  seen  from  the 
moon),  the  quantity  of  light  transmitted  through  the  atmosphere  is 
very  considerable,  and  the  moon  is  strongly  illuminated.  If,  on  the 
other  hand,  the  weather  happens  to  be  stormy  in  this  region,  the  clouds 
cut  off  nearly  all  the  light.  In  the  lunar  eclipse  of  1884  the  moon 
was  absolutely  invisible  to  the  naked  eye,  a  very  unusual  circum- 
stance on  such  an  occasion.  At  the  eclipse  of  January  28,  1888, 
Pickering  found  that  the  photographic  power  of  the  centrally  eclipsed 
moon  was  about  T-4  <yUi>  o  of  tliat  of  the  moon  when  uneclipsed. 

377.  Uses  Made  of  Lunar  Eclipses.  —  In  astronomical  importance  a 
lunar  eclipse  cannot  be  at  all  compared  with  a  solar  eclipse.     It  has  its  uses, 
however,     a.  Many  dates  in  chronology  are  fixed  by  reference  to  certain 
lunar  eclipses.     For  instance,  the  date  of  the  Christian  era  is  determined  by  a 
lunar  eclipse  which  happened  upon  the  night  before  Herod  died.     b.  Before 
better  methods  were  devised,  lunar  eclipses  were  made  use  of  to  some  extent 
in  determining  the  longitude.     Unfortunately,  as  has  been  said  (Art.  119), 
it  is  impossible  to  note  the  critical  instants  with  any  degree  of  accuracy,  on 
account  of  the  indefiniteness  of  the  earth's  shadow,     c.  The  study  of  the 
spectrum  of  the  eclipsed  moon  gives  us  some  data  as  to  the  constitution  of 
our  own  atmosphere.     We  are  thus  enabled   to  examine  light  which  has 
passed  through  a  greater  thickness  of  air  than  is  obtainable  in  any  other 
way.     d.  The  study  of  the  heat  radiated  by  the  moon  during  the  different 
phases  of  an  eclipse  gives  us  some  important  information  as  to  the  absorb- 
ing power  and  temperature  of  its  surface.     Observations  have  been  made  at 
Lord  Rosse's  observatory  of  all  the  recent  lunar  eclipses,  with  this  end  in 
view.1     e.  Finally,  at  the  time  when  the  moon  is  eclipsed,  it  is  possible  to 
observe  its  passage  over  small  stars  which  cannot  be  seen  at  all  when  near 
the  moon  except  at  such  a  time.     Observations  of  these  star  occultations 
made  at  different  parts  of  the  earth  furnish  the  best  possible  data  for  com- 
puting the  dimensions  of  the  moon,  its  parallax,  and  for  determining  its 
precise  position  in  its  orbit  at  the  time  of  observation.     The  eclipses  of  the 
last  few  years  have  been  very  carefully  observed  in  this  way  by  concert 
between  the  different  leading  observatories. 

378.  Computation  of  a  Lunar  Eclipse.  —  Since  all  the  phases  of 
a  lunar  eclipse  are  seen  everywhere  at  the  same  absolute  instant 
wherever  the  moon  is  above  the  horizon,  it  follows  that  a  single  com- 
putation giving  the  Greenwich  times  of  the  different  phenomena  is 
all  that  is  needed,  and  can  be  made  and  published  once  for  all.     Each 
observer  has  merely  to  correct  the  predicted  time  by  simply  adding 

1  See  Art.  399*  at  end  of  chapter. 


ECLIPSES   OF  THE   SUN.  255 

or  subtracting  his  longitude  from  Greenwich  in    order  to    get  the 
true  local  time.     The  computation  is  very  simple. 

The  method  of  projecting  and  calculating  a  lunar  eclipse  is  given  in  the 
Appendix  (Art.  1004). 

ECLIPSES   OF  THE   SUN. 

379.  Dimensions  of  the  Moon's  Shadow.  —  By  the  same  method 
as  that  used  for  the  shadow  of  the  earth  (merely  substituting  in  the 
formulae  the  radius  of  the  moon  for  that  of  the  earth)  ,  we  find  that 
the  length  of  the  moon's  shadow  at  any  time  is  ^L_  of  its  distance 
from  the  sun,  and  at  new  moon  averages  232,150  miles.     It  varies 
not  quite  4000  miles  each  way,  and  so  ranges  from  236,050  miles  to 
228,300.     The  semi-angle  of  the  moon's  shadow  is  practically  equal 
to  the  semi-diameter  of  the  sun  seen  at  the  earth,  or  very  nearly  16r. 

380.  The  Moon's  Shadow  on  the  Earth's  Surface.  —  Since  the 
mean  length  of  the  shadow  is  less  than  the  mean  distance  of  the 
moon  from  the  earth  (which  is  238,800  miles),  it  is  obvious  that 
on  the  average  it  will   not  reach  to  the  earth.     On  account  of  the 

B 
^\  --  -  —  ^^^l— 

M 


__ 

(T  --  "  »  -  >  to  Sun 

FIG.  129.  —  The  Moon's  Shadow  on  the  Earth. 

eccentricity  of  the  moon's  orbit  however,  our  satellite  is  much  of 
the  time  considerably  nearer  than  this  mean  distance,  and  may  come 
within  221,600  miles  from  the  earth's  centre,  or  about  217,650 
miles  from  its  surface.  The  shadow,  also,  under  favorable  circum- 
stances, may  have  a  length  of  236,050  miles.  Its  point  may  there- 
fore at  times  extend  nearly  18,400  miles  beyond  the  earth's  surface. 
The  cross-section  of  the  shadow  where  the  earth's  surface  cuts  it 
(at  o  in  Fig.  129)  will  then  be  167  miles.  This  is  the  largest  value 
possible. 

Of  course,  if  the  shadow  strikes  obliquely  on  the  surface  of  the  earth,  as 
it  must  except  when  the  moon  is  in  the  zenith,  the  shadow  spot  will  be  oval 
instead  of  circular,  and  the  length  of  the  oval  along  the  earth's  surface  may 
much  exceed  the  true  cross-section  of  the  shadow. 

381.  The  "Negative  "  Shadow.  —  Since  the  distance  of  the  moon 
may  be  as  great  as  252,970  miles  from  the  earth's  centre,  or  nearly 


256  ECLIPSES. 

249,000  miles  from  its  surface,  while  the  shadow  may  be  as  short  as 
228,300  miles,  we  may  have  the  state  of  things  indicated  by  placing 
the  earth  at  B  in  the  figure.  The  vertex  of  the  shadow,  F",  will  then 
fall  20,700  miles  short  of  the  surface,  and  the  cross-section  of  the 
"  shadow  produced "  will  have  a  diameter  of  196  miles  where  the 
earth's  surface  cuts  it.  When  the  shadow  falls  near  the  edge  of  the 
earth,  this  cross-section  may  be  as  great  as  230  miles.  The  shadow- 
spot  which  is  formed  by  the  intersection  of  the  produced  shadow- 
cone  with  the  earth's  surface  is  sometimes  called  the  negative  shadow. 

382.  Total  and  Annular  Eclipses. — To  an  observer  Within  the 
true  shadow  cone,  that  is,  between  V  and  the  moon  in  Fig.  129,  the 
sun  will  be  totally  eclipsed ;  but  an  observer  in  the  produced  cone 
beyond  V  will  see  the  moon  projected  on  the  sun,  leaving  an  un- 
eclipsed  ring  around  it.     He  will  have  what  is  called  an  annular 
eclipse.    These  annular  eclipses  are  considerably  more  frequent  than 
total  eclipses  —  nearly  in  the  ratio  of  3  to  2. 

383.  The  Penumbra  and  Partial  Eclipses. — The  penumbra  can 
easily  be  shown  to  have  a  diameter  on  the  line  CD  (Fig.  129)  of  very 
nearly  twice  the  moon's  diameter.1  We  may  take  it  as  having  an  aver- 
age diameter  at  this  point  of  4400  miles ;  but  as  the  earth  is  often 


C         F 
FIG.  130.  — Width  of  the  Penumbra  of  the  Moon's  Shadow. 

beyond  V,  its  cross-section  at  the  earth  is  sometimes  as  much  as  4800 
miles.  An  observer  situated  within  the  penumbra  observes  a  partial 
eclipse :  if  he  is  near  the  shadow  cone,  the  sun  will  be  mostly  covered 
by  the  moon  ;  but  if  near  the  outer  limit  of  the  penumbra,  the  moon 
will  only  slightly  encroach  on  the  sun's  disc.  While,  therefore,  total 
and  annular  eclipses  are  visible  as  such  only  by  an  observer  within 
the  narrow  path  traversed  by  the  shadow-spot,  the  same  eclipse 
will  be  visible  as  a  partial  one  everywhere  within  2000  miles  on 

1  Because  the  angle  DM V  (Fig.  129)  is  the  angular  diameter  of  the  sun  as 
seen  from  Jlf,  and  this  is  nearly  equal  to  the  moon's  diameter  seen  from  the 
earth,  i.e..  about  31'. 


VELOCITY    OF    SHADOW    AND   DURATION   OF    ECLIPSES.     257 

either  side  of  the  shadow  path  ;  and  the  2000  miles  is  to  be  reck- 
oned perpendicularly  to  the  axis  of  the  shadow.  When,  for  instance, 
the  penumbra  falls,  as  shown  in  Fig.  130,  the  distance  BC  measured 
along  the  earth's  surface  will  be  over  3000  miles,  although  BF  is 
only  2000. 

384.     Velocity  of  the  Shadow  and  Duration  of  Eclipses.  —  The 

moon  advances  along  its  orbit  very  nearly  2100  miles  an  hour,  and 
were  it  not  for  the  earth's  rotation  this  is  the  rate  at  which  the 
shadow  would  pass  the  observer.  The  earth,  however,  is  rotating 
towards  the  east  in  the  same  general  direction  as  that  in  which  the 
shadow  moves,  and  its  surface  moves  at  the  rate  of  about  1040  miles 


FIG.  131.  —  Track  of  the  Moon's  Shadow,  Eclipse  of  July,  1878. 

an  hour  at  the  equator.  An  observer, "there fore,  on  the  earth's 
equator,  with  the  moon  near  the  zenith,  would,  be  passed  by  the 
shadow  with  a  speed  of  about  1060  miles  per  hour  (2100  — 1040) ; 
and  this  is  its  slowest  velocity,  which  is  about  equal  to  that  of  a 
cannon-ball. 

In  higher  latitudes,  where  the  velocity  of  the  earth's  rotation  is  less, 
the  relative  speed  of  the  shadow  is  higher ;  and  where  the  shadow  falls 
very  obliquely,  as  it  does  when  an  eclipse  occurs  near  sunrise  or  sun- 


258 


ECLIPSES. 


set,  the  advance  of  the  shadow  along  the  earth's  surface  may  become 
exceedingly  swift,  —  as  great  as  4000  or  5000  miles  per  hour.  Fig. 
131,  which  we  owe  to  the  courtesy  of  the  publishers  of  Laiigley's 
"New  Astronomy,"  shows  the  track  of  the  moon's  shadow  during 
the  eclipse  of  July  29, 1878. 

385.  Duration  of  an  Eclipse,  —  A  total  eclipse  of  the  sun  observed 
at  a  station  near  the  equator  under  the  most  favorable  conditions 
possible   (the  shadow-spot  having  its  maximum  diameter  of   167 
miles),  may  continue  total  for  seven  'minutes  and  fifty-eight  seconds. 
In  latitude  40°  the  duration  of  totality  can  barely  equal  six  and  one 
quarter  minutes.     The  greatest  possible  excess  of  the  radius  of  the 
moon  over  that  of  the  sun  is  only  1'  19". 

An  annular  eclipse  may  last  for  12m  24s  at  the  equator.  The 
maximum  width  of  the  ring  of  the  sun  visible  around  the  moon 
is  1'  37". 

In  the  observation  of  an  eclipse  four  "contacts"  are  noted:  the 
first,  when  the  edge  of  the  moon  first  touches  the  edge  of  the  sun ; 
the  second,  when  the  eclipse  becomes  total  or  annular ;  the  third,  at 
the  cessation  of  the  total  or  annular  phase ;  and  the  fourth,  when 
the  moon  finally  leaves  the  disc  of  the  sun.  From  the  first  contact 
to  the  fourth  the  duration  may  be  a  little  over  four  hours. 

386.  Tire  Solar  Ecliptic  Limits.  —  It  is  necessary,  in  order  to  have 
an  eclipse  of  the  sun,  that  the  moon  should  encroach  on  the  cone 
ACBD  (Fig.  132),  which  envelops  earth  and  sun.     In  this  case  the 
"true"  angular  distance  between  the  centres  of  the  sun  and  moon 


D 


FIG.  132.  —  Solar  Ecliptic  Limits. 


—  that  is,  their  distance  as  seen  from  the  centre  of  the  earth  —  would 
be  the  angle  MES  in  the  figure.  This  is  made  up  of  three  angles  : 
MEF,  which  equals  the  moon's  semi-diameter,  S';  AES,  the  sun's 
semi-diameter,  S\  and  FEA.  This  latter  angle  is  equal  to  the  differ- 


PHENOMENA   OF   A    SOLAR   ECLIPSE. 


259 


enee  between  CFE  and  FAE.  CFE  is  the  moon's  horizontal  paral- 
lax (the  semi-diameter  of  the  earth  seen  from  the  moon),  and  FAE 
or  CAE  is  the  sun's  parallax.  FEA,  therefore,  equals  P—p\  and 
the  whole  angle  MES  equals  S  +  S'  +  P—p.  This  angle  may  range 
from  1°  34'  13"  to  1°  24'  19",  according  to  the  changing  distances  1  of 
the  sun  and  moon  from  the  earth. 

The  corresponding  distances  of  the  sun  from  the  node,  calculated  in 
the  same  way  as  the  lunar  ecliptic  limits  (taking  the  maximum  incli- 
nation of  the  moon's  orbit  as  5°  19'  and  the  minimum  as  4°  57', 
according  to  Neison),  give  18°  31f  and  15°  21'  for  the  major  and 
miuor  ecliptic  limits. 

In  order  that  an  eclipse  may  be  central  (total  or  annular)  at  any 
part  of  the  earth,  it  is  necessary  that  the  moon  should  lie  wholly 
inside  the  cone  ACBD,  as  at  M1.  In  this  case  the  angle  M'ES  will 
be  S  —  S'  +  P— p,  and  the  corresponding  major  and  minor  central 
ecliptic  limits  come  out  11°  50'  and  9°  55'. 

387.  Phenomena  of  a  Solar  Eclipse.  —  There  is  nothing  of  special 
interest  until  the  sun  is  mostly  covered,  though  before  that  time  the 
shadows  cast  by  foliage  begin  to  look  peculiar.  The  light  shining 
through  every  small  interstice  among  the  leaves,  instead  of  forming  a 
little  circle  on  the  earth,  makes  a  little  crescent  —  an  image  of  the 
partly  covered  sun. 

Some  ten  minutes  before  totality  the  darkness  begins  to  be  felt,  and 
the  remaining  light,  coming  as  it  does  from  the  edge  of  the  sun  only, 
is  much  altered  in  quality,  producing  an  effect  very  like  that  of 
a  calcium  light  rather  than  sunshine.  Animals  are  perplexed,  and 
birds  go  to  roost.  The  temperature  falls  a  few  degrees,  and  some- 
times dew  appears. 


1  We  give  herewith  in  a  table  the  different  quantities  which  determine  the 
dimensions  of  the  shadows  of  the  earth  and  moon,  as  well  as  the  ecliptic  limits 
and  the  duration  of  eclipses.  . 


Greatest. 

Least. 

Mean. 

Apparent  semi  -diameter  of  sun    . 

16'  18" 

15'  46" 

16'  02" 

Apparent  semi-diameter  of  moon 

16'  46" 

14'  42" 

15'  34" 

Horizontal  parallax  of  the  sun     . 

8".  95 

8".65 

8".80 

Horizontal  parallax  of  the  moon 

61'  28'' 

53'  55" 

57'  02" 

Inclination  of  moon's  orbit 

5°  19' 

4°  57' 

5°    8'  43" 

Sun's  radius,  433200  miles;  earth's  (mean),  3956;  moon's,  1081.5. 


260  ECLIPSES. 

In  a  few  moments,  if  the  observer  is  so  situated  that  his  view 
commands  a  distant  western  horizon,  the  moon's  shadow  is  seen 
coming  much  like  a  heavy  thunder-storm.  It  advances  with  almost 
terrifying  swiftness  until  it  envelops  him. 

For  a  moment  the  air  appears  to  quiver,  and  on  every  white  sur- 
face bands  or  "fringes"  alternately  light  and  dark,  appear.  They 
are  a  few  inches  wide  and  from  a  foot  to  three  feet  apart,  and  on 
the  whole  seem  to  be  parallel  to  the  edge  of  the  shadow.  Probably 
they  travel  with  the  wind;  but  observations  on  this  point  are  as  yet 
hardly  decisive.  The  phenomenon  is  not  fully  explained,  but  is 
probably  due  to  irregular  atmospheric  refraction  of  the  light  com- 
ing from  the  indefinitely  narrow  strip  of  the  sun's  limb  on  the  point 
of  disappearing. 

388.  Appearance  of  the  Corona  and  Prominences.  —  As  soon  as 
the  shadow  arrives,  and  sometimes  a  little  before  it,  the  corona  and 
prominences  become  visible.     The  stars  of  the  first  three  magnitudes 
make  their  appearance  at  the  same  time. 

The  suddenness  with  which  the  darkness  descends  upon  the  ob- 
server is  exceedingly  striking ;  the  sun  is  so  brilliant  that  even  the 
small  portion  which  remains  visible  up  to  within  a  very  few  seconds 
of  the  time  of  totality  so  dazzles  the  eye  that  it  is  not  prepared 
for  the  sudden  transition.  In  a  few  moments,  however,  the  vision 
becomes  accustomed  to  the  changed  circumstances,  and  it  is  then 
found  that  the  darkness  is  not  really  very  intense.  If  the  totality 
is  of  short  duration,  —  that  is,  if  the  diameter  of  the  moon  exceeds 
that  of  the  sun  by  less  than  a  minute  of  arc,  —  the  lower  parts  of 
the  corona  and  chromosphere,  which  are  very  brilliant,  give  a  light 
at  least  three  or  four  times  as  great  as  that  of  the  full  moon.  Since 
the  shadow  also  in  such  a  case  is  of  small  diameter,  a  large  quantity 
of  light  is  sent  in  from  the  surrounding  air,  where  thirty  or  forty 
miles  away  the  sun  is  still  shining ;  and  what  may  seem  remarkable, 
this  intrusion  of  outside  light  is  greatest  when  the  sky  is  clouded. 
In'  such  an  eclipse  there  is  not  much  difficulty  in  reading  an  ordinary 
watch-face.  In  an  eclipse  of  long  duration  (say  five  or  six  minutes) 
it  is  much  darker,  and  lanterns  are  necessary. 

389.  Observations  of  an  Eclipse.  —  A  total  solar  eclipse   offers  an 
opportunity  of  making  an  immense  number  of  observations  of  great  impor- 
tance which  are  possible  at  no  other  time,  besides  certain  others  which  can 
also  be  made  during  a  partial  eclipse.     We  mention  (a)  Times  of  the  four 
contacts,  and  direction  of  the  line  joining  the  cusps  during  the  partial  phases. 


CALCULATION  OF  A   SOLAR   ECLIPSE.  261 

These  observations  determine  accurately  the  relative  position  of  the  sun  and 
moon  at  the  time,  and  so  furnish  the  means  for  correcting  the  tables  of  their 
motion.  (6)  The  search  for  intra-mercurial  planets.  It  has  been  thought 
likely  that  there  may  be  one  or  more  planets  between  the  orbit  of  Mercury 
and  the  sun,  and  during  a  total  eclipse  they  would  become  visible,  if  ever. 
On  the  whole,  however,  the  observations,  so  far  made,  negative  the  existence 
of  any  body  of  considerable  size  in  this  region,  though  in  1878,  Professor 
Watson  and  Mr.  Swift,  it  was  thought,  had  discovered  one,  if  not  two,  such 
planets,  (c)  Observations  on  the  fringes,  which  have  been  described  as  show- 
ing themselves  at  the  commencement  of  totality.  Probably  the  phenomenon 
is  merely  atmospheric  and  of  little  importance,  but  it  is  not  yet  sufficiently 
understood,  (d)  Photometric  measurements  of  the  intensity  of  the  light  at 
different  stages  of  the  eclipse  and  during  totality,  (e)  Telescopic  observations 
of  the  details  of  the  prominences  and  of  the  corona.  (/)  Spectroscopic  observa- 
tions, both  visual  and  photographic,  upon  the  spectra  of  the  lower  atmos- 
phere of  the  sun,  the  prominences,  and  the  corona.  (</)  Observations  with  the 
polariscope  upon  the  polarization  of  the  light  of  the  corona,  for  the  purpose 
of  determining  the  relation  between  the  reflected  and  intrinsic  light,  and 
perhaps  the  size  of  the  reflecting  particles  which  are  distributed  through  the 
corona,  (h}  Photography,  both  of  the  partial  phases  and  of  the  corona. 

390.  Calculation  of  a  Solar  Eclipse.  —  The  calculation  of  a  solar 
eclipse  cannot  be  dealt  with  in  any  such  summary  way  as  that  of  a  lunar 
eclipse,  owing  to  the  moon's  parallax,  which  makes  the  times  of  contact  and 
other  phenomena  different  at  every  different  station.     The  moon's  apparent 
path  in  the  sky,  relative  to  the  centre  of  the  sun,  is  not  even  a  portion  of  a 
great  circle,  nor  is  it  described  with  a  uniform  velocity.     Moreover,  since 
the  phenomena  of  a  solar  eclipse  admit  of  very  accurate  observations,  it  is 
necessary  to  take  account  of  numerous  little  details  which  are  of  no  import- 
ance in  a  lunar  eclipse. 

Certain  data  for  each  solar  eclipse  hold  good  wherever  the  observer  may 
be.  These  are  calculated  beforehand  and  published  in  the  nautical  alma- 
nacs ;  and  from  them,  with  the  knowledge  of  his  geographical  position,  the 
observer  can  work  out  the  results  for  his  own  station.  But  the  calculations 
are  somewhat  complicated  and  lie  beyond  our  scope.  The  reader  is  referred 
to  any  work  on  practical  astronomy;  Chauvenet  and  Loomis  treat  the  mat- 
ter very  fully.  Th.  von  Oppolzer,  lately  deceased,  published  at  Vienna  in 
1887  a  most  remarkable  and  monumental  work  entitled  "Canon  der  Finster- 
nisse"  ("Canon  of  Eclipses"),  containing  the  approximate  elements  of  all 
eclipses  (8000  solar  and  5200  lunar)  between  the  years  1207  B.C.  and  2162 
A.D.,  with  charts  showing  the  approximate  track  of  the  moon's  shadow  for 
all  annular  and  total  eclipses  of  the  sun. 

391.  Number  of  Eclipses  in  a  Year.  — The  least  possible  number 
is  two,  both  central  eclipses  of  the  sun.    The  largest  possible  number 


262  ECLIPSES. 

is  seven,  five  of  the  sun  and  two  of  the  moon.  The  eclipses  each  year 
happen  at  two  seasons  (which  may  be  called  the  "eclipse  months"), 
half  a  year  apart  —  about  the  times,  of  course,  at  which  the  sun  in  its 
annual  path  crosses  the  two  nodes  of  the  moon's  orbit.  If  these  nodes 
were  stationary,  the  eclipse  months  would  be  always  the  same  ;  but 
because  the  nodes  retrograde  around  the  ecliptic  once  in  about  nine- 
teen years,  the  eclipse  months  are  continually  changing.  The  time 
required  by  the  sun  in  passing  around  from  a  node  to  the  same 
node  again  is  346.62  days,  which  is  sometimes  called  the  "  eclipse 
year." 

392.  Number  of  Lunar  Eclipses.  — Representing  the  ecliptic  by  a 
circle  (Fig.  133)  with  the  two  opposite  nodes  A  and  a,  it  is  easy  to  see 
first,  that  there  can  be  but  two  lunar  eclipses  in  a  year  (omitting  for 
a  moment  one  exceptional  case).  The  major  lunar  ecliptic  limit  is 
12°  15f ;  hence  there  is  only  a  space  of  twice  that  amount,  or  24°  30', 

between  L  and  L',  at  each  "  node 
month,"  within  which  the  occur- 
rence of  a  full  moon  might  give 
a  lunar  eclipse.  Now,  in  a  syn- 
odic month  the  sun  moves  along 
the  ecliptic  29°  6',  while  the  node 
moves  in  the  opposite  direction 
1°  31',  giving  the  relative  motion 
of  the  sun  referred  to  the  node 
equal  to  30°  37';  i.e.,  the  full- 
moon  points  on  the  circle  would  fall 
at  a  distance  of  30°  37'  from  each 

FIG.  133.  - Numbe^TEclipses  Annually.          °ther'      On}J  OUe  f ul1  m°°n>  ther6- 

fore,  can  possibly  occur  within 
the  lunar  ecliptic  limits  each  time  that  the  sun  passes  the  node. 

Since  the  minor  ecliptic  limit  for  the  moon  is  only  9°  30',  it  may 
easily  happen  that  neither  of  the  full  moons  which  occur  nearest  to 
the  time  when  the  sun  is  at  the  node  will  fall  within  the  limit.  There 
are  accordingly  many  years  which  have  no  lunar  eclipses. 

Three  lunar  eclipses,  however,  may  possibly  happen  in  one  calendar 
year  in  the  following  way.  Suppose  the  first  eclipse  occurs  about 
Jan.  1 ,  the  sun  passing  the  node  about  that  time  ;  the  second  may  then 
happen  about  June  25  at  the  other  node,  a.  The  first  node,  A,  will 
run  back  during  the  year,  so  that  the  sun  will  encounter  it  again  about 
Dec.  13  at  A',  and  thus  a  third  eclipse  may  occur  in  December  of 


FREQUENCY    OF    ECLIPSES.  263 

the  same  year.     This  occurred  last  in  1852,  and  will  happen  again 
in  1898  and  1917. 

393.  Number  of  Solar  Eclipses.  —  Considering  now  solar  eclipses, 
we  find  that  there  must  inevitably  be  two.     Twice  the  minor  limit 
(Art.  386)  of  a  solar  eclipse  (15°  21')  is  30°  42',  which  is  more  than 
the  sun's  whole  motion  in  a  month.     One  new  moon,  at  least,  there- 
fore, must  fall   within   the   limiting  distance   of   the  node,  and  two 
may  do  so,  since  in  the  figure,  SS1  is  always  greater  than  the  distance 
between  the  points  occupied  by  two  successive  new  moons. 

If  the  two  new  moons  in  the  two  eclipse  months  happen  to  fall 
very  near  a  node,  the  two  fall  moons,  a  fortnight  earlier  and  later, 
will  both  be  very  likely  to  fall  outside  the  lunar  limit.  In  that  case 
the  year  will  have  only  two  eclipses,  both  solar  and  both  central ;  i.e., 
either  total  or  annular ;  as  in  1904  and  1917. 

Again,  if  in  any  year  two  full  moons  occur  when  the  sun  is  very  near 
the  node,  then  since  the  major  solar  limit  is  18°  31',  it  may  happen, 
and  often  does,  that  there  will  be  two  partial  solar  eclipses,  one  a 
fortnight  before,  the  other  a  fortnight  after,  each  of  the  lunar  eclipses, 
and  so  the  year  will  have  three  eclipses  in  each  eclipse  month  —  six 
eclipses  in  all,  two  lunar  and  four  solar.  A  fftli  solar  eclipse  may 
also  come  in  near  the  end  of  the  year,  if  the  node  was  passed  about 
Jan.  15,  in  the  same  way  that  sometimes  happens  with  a  lunar  eclipse  : 
the  year  will  then  have  seven  eclipses.  This  was  the  case  in  1823. 
and  will  next  happen  in  1935.  The  most  usual  number  of  eclipses 
is  four  or  five. 

394.  Relative  Frequency  of   Solar  and  Lunar  Eclipses.  —  Al- 
though, taking  the  whole  earth  into  account,  the   solar  eclipses  are 
the  most  numerous,  about  in  the  proportion  of  four  to  three,  it  is 
not  so  with  the  eclipses  visible  at  any  given  place.     A  solar  eclipse 
can  be  seen  only  from  a  limited  portion  of  the  globe,  while  a  lunar 
eclipse  is  visible  over  considerably  more  than  half  the  earth,  either  at 
the  beginning  or  end,  if  not  throughout  its  whole  duration ;  and  this 
more  than  reverses  the  proportion  between  lunar  and  solar  eclipses 
for  any  given  station. 

395.  Recurrence  of  Eclipses,  and  the  Saros. — It  is  not  known  how 
early  it  was  discovered  that  eclipses  recur  at  a  regular  interval  of 
eighteen  years  and  eleven  and  one-third  days  (ten  and  one-third  days, 
if  there  happen  to  be  five  leap  years  in  the  interval)  ;  but  the  Chaldeans 


264  ECLIPSES. 

knew  the  period  very  well,  and  called  it  the  Saros  (which  means 
"restitution"  or  "repetition"),  and  used  it  in  predicting  the  recur- 
rence of  these  phenomena.  It  is  a  period  of  223  synodic  months, 
which  is  almost  exactly  equal  to  nineteen  eclipse  years.  The  eclipse 
year  is  346d.6201,  and  nineteen  of  them  equal  6585d.78,  while  223 
months  equal  6585d.32. 

The  difference  is  only  y4^  of  a  day  (about  11  hours)  in  which  time 
the  sun  moves  28'.  If,  therefore,  an  eclipse  should  occur  to-day  at 
new  moon,  with  the  sun  exactly  at  the  node,  then  after  223  months 
(18  years  11  days)  a  new  moon  will  occur  again  with  the  sun  only  28' 
west  of  the  node  ;  so  that  the  circumstances  of  the  first  eclipse  will 
be  pretty  nearly  repeated.  It  would  however  occur  about  eight  hours 
of  longitude  further  west  on  the  earth's  surface,  since  the  223  months 
exceed  the  even  6585  days  by  T\2¥  of  a  day,  or  7h  42m. 


As  an  example,  the  four  eclipses  of  1878  occurred  as  follows  : 
February  2,  solar,  annular  ;  February  17,  lunar,  partial  ;  July  29, 
solar,  total  ;  and  August  12,  lunar,  partial.  In  1896  the  correspond- 
ing eclipses  were  :  February  13,  solar,  annular  ;  February  28,  lunar, 
partial  ;  August  9,  solar,  total  ;  and  August  23,  lunar,  partial. 

396.  Number  of  Recurrences  of  a  Given  Eclipse.  —  It  is  usual  to 
speak  of  eclipses  recurring  at  this  regular  interval  as  "repetitions"  of  one 
and  the  same  eclipse.  Thus,  the  total  solar  eclipses  of  April  1846,  May  1864, 
May  1882,  May  1900,  June  1918,  June  1936,  June  1954,  July  -1972,  and 
August  2008  are  for  many  purposes  considered  as  mere  recurrences  of  one  and 
the  same  phenomenon.  A  lunar  eclipse  is  usually  thus  "repeated"  48  or  49 
times.  Beginning  as  a  very  small  partial  eclipse,  with  the  sun  about  12° 
east  of  the  node,  it  will  be  a  little  larger  at  its  next  occurrence  eighteen 
years  later  ;  and  after  13  or  14  repetitions  the  sun  will  have  come  so  near 
the  node  that  the  eclipse  will  have  become  total.  It  will  then  be  repeated 
as  a  total  eclipse  22  or  23  times,  after  which  it  will  become  partial  again 
with  the  sun  west  of  the  node,  and  after  13  more  returns  as  a  partial  eclipse 
will  finally  dwindle  away  and  disappear,  having  thus  recurred  regularly  once 
in  every  223  months  during  an  interval  of  865|  years. 

The  same  thing  happens  with  the  solar  eclipses,  only  since  the  solar 
ecliptic  limit  is  larger  than  the  lunar,  a  solar  eclipse  has  from  68  to  75  re- 
turns, occupying  some  1260  years.  Of  these  about  25  are  only  partial 
eclipses,  the  sun  being  so  near  the  ecliptic  limit  that  the  axis  of  the  shadow 
does  not  reach  the  earth  at  all.  The  45  eclipses  in  the  middle  of  the  period 
are  central  somewhere  or  other  on  the  earth,  about  18  of  them  being  total, 
and  about  27  annular.  These  numbers  vary  somewhat,  however,  in  differ- 
ent cases. 


NUMBER    OF   ECLIPSES   IN    A    SINGLE   SAROS.  265 

397.  It  is  to  be  noticed  that  the  Saros  exhibits  not  only  a  close 
commensurability  of  the  synodic  months  with  the  eclipse  years,  but 
also  with  the  nodical1  and  anomalistic  months :  242  nodical  months 
equal  6585.357   days;  239  anomalistic  months  equal  6585.549  days. 
This  last  coincidence  is  important.     The  moon  at  the  end   of   the 
Saros    of    223  months  not  only  returns  very  closely  to  its   original 
position  with  respect  to  the  sun  and  the  node,  but  also  with  respect  to 
the  line  of  apsides  of  its  orbit.     If  it  was  at  perigee  originally,  it  will 
again  be  within  five  hours  of  perigee  at  the  end  of  the  Saros.     If  this 
were  not  so,  the  time  of  the  eclipse  might  be  displaced  several  hours 
by  the  perturbations  of  the  moon's  motion,  to  be  considered  later, 
in  Chap.  XII. 

398.  Number  of  Eclipses  in  a  Single  Saros.  —  The  total  number  is 
usually  about  seventy,  varying  two  or  three  one  way  or  the  other,  as 
new  eclipses  come  in  at  the  eastern  limit  and  go  out  at  the  western. 
Of  the  70,  29  are  usually  lunar  and  41  solar ;  and  of  the  solar,  27  are 
central,  17  being  annular  and  10  total.     (These  numbers  are  necessarily 
only  approximate.)     It  appears,  therefore,  that  total  solar  eclipses, 
somewhere  or  other  on  the  earth,  are  not  very  rare,  there  being  about 
ten  in  eighteen  }'ears.     Since,  however,  the  shadow  track  averages 
less  than  100  miles  in  width,  each  total  eclipse  is  visible,  as  total, 
over  only  a  very  small  fraction  of  the  earth's  whole  surface  —  about 
Y  J-Q  in  the  mean.     This  gives  about  one  total  eclipse  in  360  years,  in 
the  long  run,  at  any  given  station. 

The  total  solar  eclipses  visible  in  the  United  States  during  the  nineteenth 
century  have  been  the  following  :  — 

June  16,  1806,  in  New  York  and  New  England,  duration  4|  minutes; 
Nov.  30,  1834,  in  Arkansas,  Missouri,  Alabama,  and  Georgia,  duration  2 
minutes;  July  18,  1860,  in  Washington  Territory  and  Labrador^  3  minutes; 
Aug.  7, 1869,  in  Iowa,  Illinois,  Kentucky,  North  Carolina,  2|  minutes;  July  29, 
1878,  in  Wyoming,  Colorado,  Texas,  2  \  minutes ;  Jan.  11, 1880,  in  California, 
duration  32  seconds;  Jan.  1,  1889,  in  California  and  Montana,  2£  minutes; 
On  the  morning  of  May  28,  1900,  the  moon's  shadow  crossed  the  country 
from  Texas  to  Virginia,  the  totality  lasting  in  Virginia  about  two  minutes. 

Total  eclipses  visible  in  the  United  States  occur  during  this  century 
in  1918,  1923,  1925,  1945,  1954,  1979,  1984,  and  1994,  according  to  Oppol- 
zer's  "  Canon." 


1  The  nodical  month  is  the  time  of  the  moon's  revolution  from  one  of  its  nodes 
to  the  same  node  again,  and  is  equal  to  27d.2l222  ;  the  anomalistic  month  is 
the  time  of  revolution  from  perigee  to  perigee  again,  and  equals  27d.55460.  See 
Arts.  454,  455.  The  nodical  month  is  also  called  the  draconitic  month. 


266  ECLIPSES. 

399.  Occultations  of  Stars.  — In  theory,  and  in  the  method  of  com- 
putation, the  occultation  of  a  star  is  precisely  like  a  solar  eclipse,  except 
that  the  shadow  of  the  moon  projected  by  a  star  is  a  cylinder  instead 
of  a  cone,  since,  compared  with  the  distance  of  the  sun,  that  of  a 
star  is  infinite :  moreover,  the  star  is  a  mere  point,  so  that  there  is 
no  sensible  penumbra.  In  other  words,  a  star  has  neither  parallax 
nor  semi-diameter,  and  these  circumstances  somewhat  simplify  the 
formulae. 

As  the  moon  moves  always  towards  the  east,  the  disappearance  of 
the  star  always  takes  place  at  the  eastern  limb,  and  the  reappearance 
at  the  western.  In  the  first  half  of  the  lunation  the  eastern  limb 
is  dark  and  invisible,  and  the  star  vanishes  without  warning.  The 
suddenness  with  which  it  vanishes  and  reappears  has  already  been 
referred  to  (Art.  255)  as  proof  of  the  non-existence  of  a  lunar  atmos- 
phere. Observations  of  this  sort  determine  the  moon's  place  with 
great  accuracy,  and  when  corresponding  observations  are  made  at 
different  places,  they  supply  one  of  the  best  possible  means  of  de- 
termining their  difference  of  longitude. 

In  some  cases  observers  have  reported  that  a  star,  instead  of  disappearing 
instantaneously  when  struck  by  the  moon's  limb  (faintly  visible  by  earth- 
shine),  has  appeared  to  cling  to  the  limb  for  a  second  or  two  before  vanish- 
ing, and  in  a  few  instances  they  have  even  reported  it  as  having  reappeared 
and  disappeared  a  second  time,  as  if  it  had  been  for  a  moment  visible  through 
a  rift  in  the  moon's  crust.  Some  of  these  anomalous  phenomena  have  been 
explained  by  the  subsequent  discovery  that  the  star  was  double,  or  had  a 
faint  companion,  but  for  the  most  part  are  probably  due  to  "  bad  definition" 
of  air  or  instrument,  or  to  physiological  causes  in  the  observer. 

399*.  (Supplementary  to  Art.  377.)  During  the  progress  of  a  lunar 
eclipse  the  heat  radiated  from  the  moon  varies  almost  exactly  with  the  light, 
so  that  when  the  totality  begins  the  heat  has  lost  full  98  per  cent  of  its 
original  amount,  and  during  the  totality  falls  off  about  one  per  cent  more. 
Then,  as  the  light  returns,  the  heat  rises  almost  as  rapidly  as  it  fell.  This 
indicates  that  the  lunar  surface  has  almost  no  power  of  "storing"  heat,  —  a 
natural  consequence  of  its  airlessness. 

A  very  singular  fact,  moreover,  is  that  after  the  eclipse  the  lunar  radiation 
does  not  for  several  hours  recover  the  value  it  had  before  the  eclipse  began.  In 


ECLIPSES.  267 

1888,  when  the  moon  left  the  penumbra,  and  was  again  receiving  unobstructed 
sunshine,  the  heat  had  risen  to  only  80  per  cent  of  the  original  value,  and 
lh  40m  later  had  gained  only  one  per  cent  more.  The  same  thing  was 
observed  in  1884.  No  explanation  as  yet  appears. 

NOTE  ON  THE  SOLAR  ECLIPSE  OF  JANUARY  22o,  1898. 

The  shadow  of  the  moon  traversed  the  continent  of  India  in  the  early 
afternoon,  striking  the  western  coast  about  150  miles  south  of  Bombay,  and 
crossing  the  Himalayas  near  Mt.  Everest.  The  weather  was  fine  all  along 
its  path,  and  the  numerous  observers  at  more  than  a  dozen  stations  were 
brilliantly  successful,  almost  without  exception.  The  duration  of  totality, 
however,  was  unfortunately  very  short,  —  hardly  two  minutes. 

It  was  too  early  when  this  chapter  went  to  press  to  give  the  final  results 
of  the  observations ;  it  is  probable  that  the  abundant  data  collected  will  soon 
be  decisive  in  respect  to  many  important  problems  of  solar  physics,  —  espe- 
cially those  which  relate  to  the  origin  of  the  Fraunhofer  lines. 

The  "  flash-spectrum"  (see  note  to  Art.  319)  was  successfully  photographed 
at  a  number  of  stations,  both  with  spectroscopes  of  the  ordinary  form  and 
with  prismatic  cameras,  and  with  a  dispersion  fully  double  that  used  by 
Mr.  Shackleton  in  1896.  The  comparison  of  these  photographs  with  those 
of  the  ordinary  solar  spectrum  made  with  similar  dispersive  power  can 
hardly  fail  to  determine  what  lines  originate  low  down  in  the  solar  atmos- 
phere, and  which  of  them,  if  any,  are  produced  only  in  its  upper  levels. 

Fully  a  hundred  photographs  of  the  corona  were  made  with  instruments 
ranging  from  telescopes  of  40-feet  focal  length  down  to  small  cameras. 
One  set  of  negatives  made  with  a  polariscopic  apparatus  shows  distinctly 
the  polarization  of  one  of  the  longest  streamers,  indicating  the  presence  of 
dust  or  mist.  The  gaseous  elements  of  the  corona  were,  however,  unusually 
faint,  and  no  advance  seems  to  have  been  made  towards  determining  more 
exactly  the  true  position  of  the  violet  rings  in  the  corona  spectrum,  referred 
to  in  Art.  329. 

The  corona  on  this  occasion  had  the  form  of  an  irregular  four-rayed  star, 
the  principal  streamers  issuing  from  the  sun-spot  zones,  while  the  equatorial 
extension  was  short  and  faint. 

The  photographs  made  during  the  eclipses  of  1900  and  1901  fully  confirm 
the  results  stated  above,  and  those  of  Mr.  Evershed  show  a  number  of  addi- 
tional coronium  and  helium  lines  in  the  ultra-violet. 


268  EXERCISES. 

EXERCISES  ON  CHAPTERS  X  AND  XI. 

1.  If  the  diameter  of  the  sun  is  decreasing  at  the  rate  of  300  feet  a 
year,  how  long  before  its  apparent  diameter  will  have  decreased  by  1"? 
(See  Art.  276.)  Ans.    7927  years. 

2.  If  the  rate  of  shrinkage  be  assumed  to  continue  uniform   (i.e.,  300 
feet  a  year,  an  improbable  assumption),  how  long  will  it  be  before  its 
diameter  is  diminished  by  one  per  cent?  Ans.    Over  150000. 

3.  How  much  would  its  density  then  be  increased? 

4.  Taking  the  calorie  as  equivalent  to  428  kilogram-metres  of  energy, 
what  weight  falling  100  metres  would  at  the  end  of  its  fall  possess  an  energy 
equal  to  that  of  the  solar  radiation  received  in  an  hour  upon  ten  square 
metres  of  the  earth's  surface,  allowing  for  a  loss  of  fifty  per  cent  absorbed 
by  the  air?  Ans^    38520  kilograms. 

5.  Assuming  (Art.  332)  that  sunlight  at  the  earth  equals  70000  times 
that  of  a  standard  candle  at  a  distance  of  one  metre,  at  what  distance  would 
the  light  of  the  sun  equal  that  of  a  2000  candle-power  electric  arc  ten  metres 
distant  ? 

-4ns.    About  59  times  the  earth's  distance  from  the  sun. 

6.  Can  an  eclipse  of  the  moon  ever  occur  in  the  daytime  ?     (Consider 
the  possible  effect  of  refraction.) 

7.  Why  cannot  there  be  an  annular  eclipse  of  the  moon  ? 

8.  Which  are  most  frequent  in  New  York,  solar  eclipses  or  lunar  ? 

9.  If  a  lunar  eclipse  has  occurred  this  year  in  August,  can  there  be  one 
in  June  of  next  year  ?  or  in  October  ?    If  not,  why  not  ? 

10.  Can  an  occultation  of  Venus  occur  during  an  eclipse  of  the  moon  ? 
Is  one  of  Jupiter  possible? 

11.  In  a  solar  eclipse  which  side  of  the  sun's  disc  is  first  touched  by  the 
moon,  the  east  or  the  west  ? 

12.  Does  the  shadow  of  the  moon  during  a  solar  eclipse  ever  travel  west- 
ward over  the  surface  of  the  earth  ?     (Consider  the  case  of  an  eclipse  within 
the  polar  circle  occurring  near  midnight.) 


EQUABLE  DESCRIPTION   OF   AREAS. 


269 


CHAPTER  XII. 


CENTRAL  FORCES:  EQUABLE  DESCRIPTION  OF  AREAS. — AREAL, 
LINEAR,  AND  ANGULAR  VELOCITIES.  —  KEPLER'S  LAWS  AND 
INFERENCES  FROM  THEM.  —  GRAVITATION  DEMONSTRATED 
BY  THE  MOON'S  MOTION.  —  CONIC  SECTIONS  AS  ORBITS.  — 

THE    PROBLEM    OF    TWO    BODIES.  THE    "  VELOCITY    FROM 

INFINITY,"  AND  ITS  RELATION  TO  THE  SPECIES  OF  ORBIT 
DESCRIBED  BY  A  BODY  MOVING  UNDER  GRAVITATION. — THE 
INTENSITY  OF  GRAVITATION. 

400.  A  MOVING  body  left  to  itself,  according  to  Newton's  first 
law  of  motion  (Physics,  p.  26),  moves  on  forever  in  a  straight  line 
with  a  uniform  velocity.     If  we  find  a  body  so  moving,  we  may, 
therefore,  infer  that  it  is  either  acted  on  by  no  force  whatever,  or, 
if  forces  are  acting  upon  it,  that  they  exactly  balance  each  other. 

It  has  been  customary  with  some  writers  to  speak  of  a  body  thus  moving 
"uniformly  in  a  straight  line"  as  actuated  by  a  "projectile  force"  a  very 
unfortunate  expression,  which  is  a  survival  of  the  Aristotelian  idea  that  rest 
is  more  "  natural "  to  matter  than  motion,  and  that  when  a  body  moves, 
some  force  must  operate  to  keep  it  moving.  The  mere  uniform  rectilinear 
motion  of  a  material  mass  in  empty  space  implies  no  action  of  a  physical 
cause,  and  demands  explanation  only  as  mere  existence  does.  Change  of 
motion,  either  in  speed  or  in  direction — this  alone  implies  force  in  operation. 

401.  If  a  body  moves  in  a  straight  line,  with  swiftness  either 
increasing  or  decreasing,  we  infer  a 

force  acting  exactly  in  the  line  of  mo- 
tion, and  accelerating  or  retarding  it. 
If  it  moves  in  a  curve,  we  know  that 
some  force  is  acting  athwart  the  line 
of  motion.  If  the  velocity  in  the  curve 
increases,  we  know  that  the  direction 
of  the  force  that  acts  is  forward,  like  FIG.  134.— Curvature  of  an  Orbit. 
ab  (Eig.  134),  making  an  angle  of  less 

than  90°  with  the  "line  of  motion"  at  (the  tangent  to  the  path  of 
the  body)  ;  and  vice  versa,  if  the  motion  of  the  body  is  retarded. 


270  CENTRAL  FORCES. 

If  the  speed  does  not  alter  at  all,  we  know  that  the  force  must  act 
along  the  line  ac,  exactly  perpendicular  to  the  line  of  motion. 

Here,  also,  we  find  many  writers,  the  older  ones  especially,  bringing  in 
the  "  projectile  force,"  and  saying  that  when  a  body  moves  in  a  curve  it  does 
so  under-  the  action  of  two  forces,  one  a  force  that  draws  it  sideways,  the 
other  the  "  projectile  force  "  directed  along  its  path.  We  repeat ;  this  "  pro- 
jectile force  "  has  no  present  existence  or  meaning  in  the  problem.  Such 
a  force  may  have  put  the  body  in  motion  long  ago,  but  its  function  has 
ceased,  and  now  we  have  only  to  do  with  the  action  of  one  single  force, — 
the  deflecting  force,  which  alters  the  direction  of  the  body's  motion.  Of 
course  it  is  not  intended  to  deny  that  the  deflecting  force  may  itself  be  the 
resultant  of  any  number  of  forces  all  acting  together ;  but  a  single  force  act- 
ing athwart  a  body's  line  of  motion  is  sufficient  to  cause  it  to  describe  a 
curvilinear  orbit,  and  from  such  an  orbit  we  can  only  infer  the  necessary 
existence  of  one  such  force. 

402.  Description  of  Areas.  —  (a)  If  a  body  is  moving  uniformly  on 
a  straight  line,  and  if  we  connect  the  points  A,  B,  C,  etc. ,  Fig.  135,  which 
it  occupies  at  the  end  of  successive  units  of  time  with  any  point  what- 
ever, as  0,  we  shall  have  a  series 
of  triangles, .  AOB,  etc.,  which 
will  all  be  equal;  since  their  bases 
AB,  BO,  etc.,  are  equal  and  on 
the  same  straight  line,  and  they 
have  a   common   vertex   at  0. 
Calling  the  line  from  A  to  0  its 
radius  vector,  and  0  the  "  cen- 

J.  135.  tre,"  we  may  say,  therefore,  that 

Degcription  of  Areas  in  Uniform  Motion.  when    a   body    is    moving   Undis- 

turbed  by  any  force  whatever, 

its  radius  vector,  from  any  centre  arbitrarily  chosen,  describes  equal 
areas  in  equal  times  around  that  centre.  The  area  enclosed  in  the 
triangle  described  by  the  radius  vector  in  a  unit  of  time  is  called 
the  body's  "  areal  (or  "areolar")  velocity,"  and  in  this  case  is 
constant. 

403.  (6)  Moreover,  any  impulse  in  the  line  of  the  radiuj^jvector., 
either  towards  or  from  the  centre,  leaves  unchanged  both  the  plane  of 
the  body's  motion  and  its  areal  velocity. 

Suppose  a  body  moving  uniformly  on  the  line  A G  (Fig.  136)  with 
such  a  velocity  that  it  describes  AB,  BO,  etc.,  in  successive  units  of 
time ;  then,  by  the  preceding  section,  the  areal  velocity  will  be  con- 


DESCRIPTION    OF    AEEAS. 


271 


stant,  and  measured  by  the  area  of  any  one  of  the  equal  triangles 
AOB,  BOC,  or  COL.  Suppose,  now,  that  at  C  the  body  receives 
an  " impulse"  directed  along  the  radius  vector  towards  0  —  a  blow, 
for  instance,  which  by  itself  would  make  the  body  describe  CK  in  a 
unit  of  time.  The  body  will  now  take  a  new  path,  which  will  carry 
it  to  the  point  D,  determined  by  constructing  the  "  parallelogram  of 


Pia.  136.  —  Description  of  Areas  under  an  Impulse  directed  towards  a  Centre. 

motions "  CKDL,  and  thus  combining  the  new  motion  CK  with  the 
former  motion  CL,  according  to  Newton's  second  law  (Physics,  p.  26). 
The  new  areal  velocity,  measured  by  the  triangle  OCD,  will  be  the 
same  as  before,  as  is  easily  shown. 

Triangle  BOC=  triangle  COL,  because  BC  =  CL,  and  0  is  their 
common  vertex. 

Also,  triangle  C  OL  =  triangle  COD,  because  they  have  the  com- 
mon base  OC,  and  their  summits  L  and  D  are  on  a  line  which  was 
drawn  parallel  to  this  base  in  constructing  the  parallelogram  of 
motions.  Hence,  triangle  BOC=  triangle  COD,  and  the  areal 
velocity  remains  unchanged. 

Also,  as  may  be  seen  by  following  out  the  same  reasoning  with 
CK1  and  CD',  the  same  result  would  hold  true  i£,tJie^  impulses  JiaxJ 
been  directed  aivay  from  0  instead  of  towards  it. 

404.     This  result  depends  entirely  on  the  fact  that  the  impulse  CK  or 

CK'  was  exactly  along  the  radius  vector  CO.     If  it  had  not  been  so,  then  in 


272  CENTRAL  FORCES. 

constructing  the  parallelogram  of  motions  to  find  the  points  D  and  7X,  we 
should  have  had  to  draw  LD  or  LD'  not  parallel  to  CO,  and  the  two  triangles 
BOC  and  COD  would  necessarily  have  been  unequal.  COD  would  be 
greater  than  BO C  if  CK  were  directed  ahead  of  the  radius  vector,  and  less 
if  behind  it. 

As  regards  the  plane  of  motion,  the  point  D  is  on  the  plane  OCL, 
because  LD  was  drawn  through  L  parallel  to  OC.  OCL  is  a  part 
of  the  plane  which  contains  the  triangles  BOC  and  AOB,  and  hence 
OGD  also  lies  in  the  same  plane. 

405.  (c)  From  this  obviously  follows  the  important  general  prop- 
osition that  when  a  body  is  moving  under  the  action  of  a  force  always 
directed  towards  or  from  a  flxed  centre,  the  radius  vector  ivill  describe 
equal  areas  in  equal  times;  and  the  path  of  the  body  will  all  lie  in  pne^ 
plane. 

Such  a  force  constantly  acting  is  simply  equivalent  to  an  indefinite 
number  of  separate  impulses.  Now  if  no  single  impulse  directed 
along  the  radius  vector  can  alter  the  areal  velocity  or  plane  of  motion, 
neither  can  a  succession  of  them.  Hence  the  proposition  follows. 

In  case  of  a  continuously  acting  force  the  orbit,  however,  will 
become  a  curve  instead  of  being  a  broken  line. 

Observe  that  this  proposition  remains  true  whether  the  force  is 
attractive  or  repulsive,  and  that  it  is  independent  of  the  law  of  the 
force ;  that  is,  the  force  may  vary  directly  with  the  distance,  or  in- 
versely as  the  square  of  the  distance,  or  as  the  logarithm  of  it,  or  in 
any  conceivable  way ;  it  may  even  be  discontinuous,  acting  only  at 
intervals  and  ceasing  between  times  :  and  still  the  law  holds  good. 

406.  Conversely,  if  a  body  moves  in  this  wayf  describing  equal 
areas  in  equal  times  around  a  point,  it  is  easily  shown  that  all  the 
forces  acting  upon  the  body  must  be  directed  toward  that  point. 

We,  however,  leave  the  demonstration  to  the  student. 

Since  the  earth  moves  very  nearly  in  this  way  in  its  orbit  around 
the  sun,  we  conclude  that  the  only  force  of  any  consequence  acting 
upon  the  earth  is  directed  towards  the  sun.  We  say,  "of  any  con- 
sequence," because  there  are  other  small  forces  which  do  slightly 
modify  the  earth's  motion,  and  prevent  it  from  exactly  fulfilling  the 
law  of  areas. 

As  a  direct  consequence  of  the  law  of  equal  areas  we  have  certain 
laws  with  respect  to  the  linear  and  angular  velocities  of  a  body  mov- 
ing under  the  action  of  a  central  force. 


LAW   OF   LINEAR  VELOCITY.  273 

407.  Law  of  Linear  Velocity.  —  Suppose  a  body  moving  under 
the  action  of  a  force  always  directed  towards  S  (Fig.  137),  and  let 
AB  be  a  portion  of  its  path  which  it  de- 
scribes in  a  second.     Draw  the  tangent 

Eb.  Regarding  the  sector  ASB  as  a 
triangle  (which  it  will  be,  sensibly,  since 
the  curvature  of  the  path  in  one  second 
will  be  veiy  small)  the  area  of  this  tri- 
angle will  be  ±(ABxSb).  Now  AB, 
fche  distance  travelled  in  a  second,  is 

the   linear  velocity  of  the  body  (called  'Q 

linear  because  it  is  measured  with  the  FIG.  137. 

same  units  as  any  other  line;  i.e.,  in       Linear  and  Angular  Velocities. 
miles  or  in  feet  per  second) ,  and  $6  is  the 

distance  from  the  centre  of  force  to  the  "  line  of  motion,"  as  the  tan- 
gent Bb  is  called.  For  Sb,  p  is  usually  written ;  hence  in  every 

2A 

part  of  the  same  orbit,  F(the  velocity  in  miles  per  second)  = — ,  and 

P 

is  inversely  proportional  to  p.  If  p  were  to  become  zero,  V  would 
become  infinite,  unless  A  were  zero  also. 

408.  Law  of  Angular  Velocity.  —  Referring  again  to  the  same 
figure,  the  area  of   ASB  is  equal  to  %  (AS  X  BS  X  sin  ASB),  or 
A  =  ^r1r2sm<j).     If  we  draw  r  to  the  middle  point  of  AB,  then  r^z 
=  i2,  nearly,    since   in   a   second   of   time   the   distance  would  not 
change  perceptibly  as  compared  with  its  whole  length.     to  will  also 
be  a  small  angle,  so  that  its  sine  will  equal  the  angle  itself  expressed 
in  radians ; 

2  A 
hence  l^co  —A,  and  <o  =  — ~. 

Now  <o  is  the  angular  velocity  of  the  body ;  that  is,  the  number  of 
"  radians"  which  it  describes  in  a  second  of  time,  as  seen  from  S, 
while  r  is  the  radius  vector. 

409.  In   every  case,  therefore,  of  motion  under  a  central  force, 
I.     The   Areal    velocity    (acres   per  second)  jiscgnstantj    II.    The 
Linear  velocity  (miles  per  second)  varies  inversely  as  the  distance  from 
the  centre  of  force  to  the  body's  line  of  motion  at  the  moment,  which 
line  of  motion  is  the  tangent  to  the  orbit  at  the  point  where  the  body 
happens  to  be ;  III.    The  Angular  velocity  (radians,  or  degrees,  per 
second)  varies  inversely  as  the  square  of  the  distance  of  the  body  from 
the  centre  of  force. 


274  CENTRAL   FORCES. 

410.  The  student  will  remember  that  it  was  found  by  observation 
that  the  sun's  angular  velocity  varies  as  the  square  of  its  apparent 
diameter,  and  from  this  (Art.  186)  the  law  of  equal  areas  was  inferred 
as  a  fact  with  respect  to  the  earth's  motion.  Newton  was  the  first 
to  point  out  that  a  body  moving  under  the  action  of  a  central  force 
must  necessarily  observe  this  law  of  areas,  and,  conversely,  that  a 
body  thus  observing  the  law  of  areas  must  necessarily  be  under  the 
control  of  a  central  force. 

„  411.  Circular  Motion.  —  In  the  case  of  a  body  moving  in  a  circle 
under  the  action  of  a  central  force,  the  force  must  be  constant;  and 
(Physics,  p.  17)  is  given  by  the  formula 


in  which  r  is  the  radius  of  the  circle  and  Fthe  velocity,  while/  is 
the  central  force  measured  as  an  "acceleration,"  in  metres  (or  feet) 
per  second;  that  is,  by  the  number  of  units  of  velocity  which  the 
force  would  generate  in  the  body  in  a  second  of  time ;  just  as  the 
force  of  gravity  is  expressed  by  writing,  g  =  9.81  metres. 

For  many  purposes  it  is  desirable  to  have  an  expression  which 
shall  substitute  for  V  (a  quantity  not  given  directly  by  observation) 
the  time  of  revolution,  t,  which  is  so  given.  Since  V  equals  the 

circumference  of  the  circle  divided  by  t,  or  — -,  we  have  at  once,  by 

O 

substituting  this  value  for  V  in  equation  (a), 

(5) 


This,  of  course,  is  merely  the  equivalent  of  equation  («),  but  is 
often  more  convenient. 

KEPLER'S    LAWS. 

412.  In  1607-1620  Kepler  discovered  as  facts,  without  an  expla- 
nation, three  laws  which  govern  the  motions  of  the  planets,  —  laws 
which  still  bear  his  name.  He  worked  them  out  from  a  discussion 
of  the  observations  which  Tycho  Brahe  had  made  through  many 
preceding  years.  The  three  laws  are  as  follows  :  — 

I.  The  orbit  of  each  planet  is  an  ellipse,  with  the  sun  in  one  of  its 
foci. 

II.  The  radius  vector  of  each  planet  describes  equal  areas  in  equal 
times. 


275 

III.  The  "  Harmonic  law,"  so-called.  The  squares  of  the  periods 
of  the  planets  are  proportional  to  the  cubes  of  their  mean  distances 
from  the  sun;  i.e.,  tf  \  t22  =  a-?  Ia23.  (See  Art.  423,  last  sentence.) 

413.  To  make  sure  that  the  student  apprehends  the  meaning  and  scope 
of  this  third  law,  we  append  a  few  simple  examples  of  its  application. 

(1)  What  would  be  the  period  of  a  planet  having  a  mean  distance  from 
the  sun  of  100  astronomical  units  ;  i.e.,  a  distance  100  times  that  of  the  earth? 

(Earth's  Dist.)*  :  (Planet's  Dist.)B  =  (Earth's  Period)2  :  (Planet's  Period)2', 
i.e.,  1s  :  1003.=  I2  (year)  :  X2  (years), 

whence,  X  -  100§  =  1000  years. 

(2)  What  would  be  the  distance  of  a  planet  having  a  period  of  125  years? 

(I)2  :  1252  =  1s  :  Xs, 
whence,  X  =  125^  =  25  (Astron.  units). 

^ 

(3)  How  long  would  a  planet  require  to  fall  to  the  sun  ? 

If  the  sun  were  collected  in  a  single  point  at  its  centre,  a  body  starting 
from  a  point  on  the  planet's  orbit  with  a  slight  side  motion,  i.e.,  motion  at 
right  angles  to  the  radius  vector,  would  describe  an  extremely  narrow  ellipse 
around  the  sun,  with  its  perihelion  just  at  the  sun,  and  the  aphelion  at  the 
starting-point.  Practically  it  would  "fall  to  the  sun,"  and  return  just  as  if 
it  had  rebounded  from  a  perfectly  elastic  surface:  the  time  of  "falling" 
would  be  just  equal  to  that  of  returning  —  the  two  making  up  the  whole 
period  of  the  body  in  the  narrow  ellipse.  Now  the  semi-major  axis  of  this 
narrow  ellipse  is  evidently  one-half  the  radius  of  the  planet's  orbit.  Hence, 
to  find  the  period  in  this  ellipse  which  is  2r  (r  being  taken  as  the  time  of 
"falling"),  we  have 

a8  :  (ia)8  =  t2  :  (2r)2,  or  1  :  *  =  t2  :  4r2  ; 

whence,  T  =  t^~-^  =  0.1768  1,  t  being  the  planet's  period. 

In  the  case  of  the  earth  r  =  365£  X  0.1768  =  64.56  days. 

(4)  What  would  be  the  period  of  a  satellite  revolving  close  to  the  earth's 
surface  ? 

(Moon's  Dist.)3  :  (Dist.  of  Satellite)5  =  (27.3  days)2  :  X2, 

or  603  :  13  =  (27.3)2  :  X2, 

whence,  Z 


60* 

(5)  How  much  would  an  increase  of  10  per  cent  in  the  earth's  distance 
from  the  sun  increase  the  length  of  the  year  ?  Ans.   56.13  days. 

(6)  What  is  the  distance  from  the  sun  of  an  asteroid  which  has  a  period 
of  3|  years?  Ans.   2.305  Astron.  units. 


276  CENTRAL   FORCES. 

414.  Many  surmises  were  made  as  to  the  physical  meaning  of 
these  laws.     More  than  one  astronomer  guessed  that  a  force  directed 
toward  the  sun,  or  emanating  from  it,  might  be  the  explanation. 
Newton  proved  it.     He  demonstrated  the  law  of  equal  areas  and  its 
converse  as  necessary  consequences  of  the  laws  of  motion.     He  also 
proved  that  if  a  body  move,  as  does  the  earth,  in  an  ellipse  having 
a  centre  of  force  at  its  focus,  then  the  force  at  different  points  in  the 
orbit  must  vary  inversely  as  the  square  of  the  distance  from  that 
centre.     And,  finally,  he  showed  that,  granting  the  harmonic  law, 
the  force  from  planet  to  planet  must  also  vary  according  to  the  same 
law  of  inverse  squares. 

415.  The  demonstration  of  this  last  proposition  for  circular  orbits  is  so 
simple  that  we  give  it,  merely  adding  (without  proof)  that  the  proposition 
is  equally  true  for  elliptical  orbits,  if  for  r  we  put  a,  the  semi-major  axis  of 
the  orbit. 

In  a  circular  orbit,  from  equation  (Z>),  (Art.  411),  we  have 


where  r  and  t  are  the  distance  and  period  of  a  planet.     In  the  same  way  the 
force  acting  upon  a  second  planet  is  found  from  the  equation 


whence,  —  =  — 

But  by  Kepler's  third  law     t2  :  t^  : :  r3  :  rx3, 
whence,  ^2  =  — L. 

Substitute  this  value  of  t^2  in  the  preceding  equation  ;  we  have 

•f  r         fir  3         r  2 

which  is  the  law  of  inverse  squares. 

416.     Conversely,  the  harmonic  law  is  just  as  easily  shown  to  be  a  neces- 
sary consequence  of  the  law  of  gravitation  in  the  case  of  circular  orbits. 
From  Art.  411,  Eq.  (&),  we  have 


CORRECTION    OF    KEPLER'S    THIRD    LAW.  277 


also,  from  the  law  of  gravitation, 


/  —  —$  M  being  the  mass  of  the  sun. 


Hence,  equating  the  two  values  of  /, 


M  r  47T2 

_  —  4_2_      nnrl     /2  —  _  -  rS 

r2  ~        t2  M 


Similarly  for  another  planet, 

47T2 
/  2  —  -  r  3 

h        Mri  ' 

Whence,  V  :  ^2  =  r3  :  rx3.  - 

The  demonstration  for  elliptical  orbits  is  a  little  more  complicated,  involv- 
ing the  "  law  of  areas."  It  is  given  in  all  works  on  Theoretical  Astronomy, 
and  may  be  found  in  Loomis's  "  Treatise  on  Astronomy,"  p.  134. 

-  417,  Correction  of  Kepler's  Third  Law,  --  The  "harmonic  law"  as 
it  stands  is  not  exactly  true,  though  the  difference  is  too  small  to  appear  in 
the  observations  which  Kepler  made  use  of  in  its  discovery.  It  would  be 
exactly  true  if  the  planets  were  mere  particles  of  matter  ;  but  as  a  planet's 
mass  is  a  sensible,  though  a  very  small  fraction  of  the  sun's  mass,  it  comes 
into  account.  The  planet  Jupiter,  for  instance,  attracts  the  sun  as  well  as  is 
attracted  by  it.  If  at  the  distance  r  Jupiter  is  drawn  towards  the  sun  by  a 

force  which  would  give  it  in  a  second  an  acceleration  expressed  by  G—^  (the 
sun's  mass  being  M  ),  then  the  sun  in  the  same  time  is  accelerated  towards 
Jupiter  by  the  quantity  G~^  (m  being  the  mass  of  Jupiter).  The  rate  at  which 

the  two  tend  to  approach  each  other  is  therefore  expressed  by  G  -  ^  —  •    Hence, 

in  discussing  the  motions  of  the  planet  Jupiter  around  the  centre  of  the  sun, 
instead  of  writing 

/=  G—  simply,  we  must  put/=  G-  —  -  —  ,  G  being  the  "constant  of  Gravi- 
tation "  (Art.  161). 

4?r2r 
But  (in  the  case  of  circular  motion)  /  =  ~~T^~* 

Hence,  we  find  GP(M+  ni)  -  47TV8  ; 


or,  as  a  proportion,  £2(3f  +  rn)  :  t^2(M  +  mx)  =  r8  :  r^, 

which  is  strictly  true  as  long  as  the  planet's  motions  are  undisturbed. 


278 


CENTRAL   FORCES. 


418.  Inferences  from  Kepler's  Laws.  —  From  Kepler's  laws  we 
are  entitled  to  infer  — 

First  (from  the  second  law),  that  the  force  which  retains  the  planets 
in  their  orbits  is  directed  towards  the  sun. 

Second  (from  the  first  law),  that  on  any  given  planet  the  force 
varies  inversely  as  the  square  of  its  distance  from  the  sun. 

Third  (from  the  harmonic  law),  that  the  force  is  the  same  for  one 
planet  as  it  would  be  for  another  in  the  same  place ;  or,  in  other 
words,  the  attracting  force  depends  only  on  the  mass  and  distance  of 
the  bodies  concerned,  and  is  wholly  independent  of  their  physical  con- 
ditions, such  as  their  temperature,  chemical  constitution,  etc.  It 
makes  no  difference  in  the  motion  of  a  planet  around  the  sun  whether 
it  be  made  of  hydrogen  or  iron,  whether  it  be  hot  or  cold. 

419.  Verification  of  "  Gravitation"  by  Means  of  the  Moon's  Mo- 

tion. —  When  the  idea  of  gravita- 
tion first  occurred  to  Newton  he 
endeavored  to  verify  it  by  com- 
paring the  force  which  keeps  the 
moon  in  her  orbit  with  the  force 
of  gravity  at  the  earth's  surface, 
reduced  in  the  proper  proportion. 
For  lack,  however,  of  an  accurate 
knowledge  of  the  earth's  dimen- 
sions,1 he  failed  at  first,  there  being 
a  discrepancy  of  about  sixteen  per 
cent.  He  had  assumed  a  degree 
to  be  exactly  sixty  miles  in  length. 
Some  years  afterward,  when  Pic- 
ard's  measure  of  the  arc  of  a  merid- 
ian in  Northern  France  had  been 
made  and  reported  to  the  Royal  Society,  making  a  degree  about  sixty- 
nine  miles  long,  he  saw  at  once  that  the  new  value  would  reconcile  the 
discrepancy ;  and  he  resumed  his  unfinished  work  and  completed  it. 

420.  At  the  earth's  surface  a  body  falls  about  193  inches  in  a 
second.     The  distance  of  the  moon  being  very  nearly  sixty  times  the 
earth's  radius^  if  gravity  really  varies  inversely  as  the  square  of  the 


FIG. 138. 

Verification  of  the  Hypothesis  of  Gravitation 
by  Means  of  the  Motion  of  the  Moon. 


1  He  \vas  long  baffled  also  by  the  difficulty  of  proving  that  the  attraction  of  a 
globe  is  the  same  as  if  its  matter  were  concentrated  at  its  centre. 


VERIFICATION   OF   GRAVITATION.  279 

distance,  a  stone  at  that  distance  from  the  earth  should  fall  —  or 


—  as  far;   that  is,  it  ought  to  fall  =  0.0535  inches, 


a  little  more  than  one-twentieth  of  an  inch.  Now  the  distance  which 
the  moon  actually  does  fall  towards  the  earth  in  a  second,  i.e.,  the 
deflection  of  its  orbit  from  a  straight  line  in  a  second  of  time,  is  easily 
found  ;  and  if  the  force  which  keeps  the  moon  in  its  orbit  is  realty  the 
same  as  that  which  makes  bodies  fall  towards  the  centre  of  the  earth, 
this  deflection  ought  to  come  out  equal  to  Oin>.0535.  Let  AE  (Fig. 

138)  be  the  distance  the  moon  travels  in  a  second  =—?  where  r  is  the 

radius  of  the  moon's  orbit,  and  t  the  number  of  seconds  in  a  month. 
Then,  since  AEF  is  a  right-angled  triangle,  we  have, 

AB:AE::AE:AF(or2r)  ; 
AE2 


whence  AB 


2r 


The  calculation  is  easy  enough,  though  the  numbers  are  rather  large. 
As  a  result  it  gives  us  AB  =  0.0534  inches,  which  is  practically  equal 
to  the  thirty-six  hundredth  part  of  193  inches. 

If  the  quantities  did  not  agree  in  amount,  the  discrepancy  would 
disprove  the  theory,  and,  as  we  have  said,  Newton  loyally  gave  it  up 
until  he  was  able  to  show  that  the  apparent  discordance  was  the  result 
of  a  mistake  in  the  original  data,  and  disappeared  when  the  data  were 
corrected.  The  agreement,  however,  does  not  establish  the  theory, 
but  only  renders  it  probable.  It  does  not  establish  it  completely, 
because  it  is  conceivable  that  the  agreement  might  be  a  case  of  acci- 
dental coincidence,  while  the  forces  might  really  differ  as  much  in 
their  nature  as  an  electrical  attraction  and  a  magnetic. 

421.  Newton  was  not  satisfied  with  merely  showing  that  the  prin- 
cipal motions  of  the  planets  and  the  moon  could  be  explained  by  the 
law  of  gravitation  ;  but  he  went  on  to  investigate  the  converse  prob- 
lem, and  to  determine  what  must  be  the  motions  necessary  under  that 
law.  He  found  that  the  orbit  of  a  body  moving  around  a  central  mass 
is  not  of  necessity  a  circle,  or  even  a  nearly  circular  ellipse  like  the 
planetary  orbits,  but  that  it  may  be  a  conic  section  of  any  eccentricity 
whatever  —  a  circle,  ellipse,  parabola,  or  even  an  hyperbola ;  but  it 
must  be  a  conic. 


280 


CENTRAL   FORCES. 


422.  For  the  benefit  of  those  of  our  readers  who  are  not  acquainted 
with  conic  sections  we  give  the  following  brief  account  of  them 
(Fig.  139)  :  — 

a.    If  a  cone  of  any  angle  be  cut  perpendicularly  to  the  axis,  the 

section  will  be  a  circle — M N  in  the 
figure. 

b.  If  it  be  cut  by  a  plane  which 
makes  with  the  axis  an  angle  greater 
than  the  semi- angle  of  the  cone,  so 
that   the  plane   of  section  cuts  com- 
pletely across  the  cone  (as  EF) ,  the 
section  is  an  ellipse;  the  circle  being 
merely  a  special  case  of  the  ellipse. 
Ellipses,  of  course,  differ  greatly  in 
form,   from  those   which    are   very 
narrow  to  the  perfect  circle. 

c.  The   parabola    is    formed    by 
cutting  the  cone  with  a  plane  parallel 
to  its  side;  i.e.,  making  with  the  axis 
an  angle  equal  to  the  semi-angle  of 
the  cone.    RPO  is  such  a  plane.    As 
all  circles  are  alike  in  form,  so  are 
all    parabolas,   whatever   the    angle 
of  the  cone  at   V  and  wherever  the 
point   P  is   taken.     If   the    cutting 

\B  plane  is  thus  situated,  then,  no 
matter  what  is  the  angle  of  the 
cone  or  the  place  where  the  cut  is 
made,  the  (complete)  curve  will 
always  be  the  same  in  shape,  though 
of  course  its  size  will  depend  upon 
a  variety  of  circumstances.  The 
statement  seems  at  first  a  little  surprising ;  but  it  is  true. 

d.  If  the  cutting  plane  makes  an  angle  with  the  axis  of  the  cone 
less  than  the  semi-angle  at  V,  so  that  the  cutting  plane  gets  continually 
deeper  and  deeper  into  the  cone,  then  the  curve  is  an  hyperbola;  so 
called,  because  the  plane  in  this  case  ''shoots  over"  (wrep  fiaXXtw) 
and  intersects  the  u  cone  produced,"  cutting  out  of  this  second  cone 
a  curve  precisely  like  the  curve  cut  from  the  original,  as  at  ITGr'IC  in 
the  figure.  The  axis  of  the  hyperbola  lies  outside  of  the  curve  itself, 
being  the  line  HIT  in  the  figure,  and  the  "centre"  of  the  curve  is 
also  outside  of  the  curve  at  the  middle  point  of  this  axis. 


FIG.  139.  —  The  Conies. 


THE    CONIC    SECTIONS. 


281 


423,  Philosophically  speaking  there  are  therefore  but  two  species 
of  conic  sections,  —  the  ellipse  and  the  hyperbola,  with  the  parabola 
for  a  partition  between  them.  (The  circle,  as  has  been  said  before, 
is  merely  a  special  case  of  the  ellipse.)  Fig.  140  will  give  the  reader 


FIG.  140.  —  The  Relation  of  the  Conies  to  Each  Other. 

perhaps  a  better  idea  of  the  nature  of  the  curves  as  drawn  on  a  plane. 
In  the  ellipse  the  sum  of  the  distances  from  the  two  foci,  FN-\-  F'N, 
equals  the  major  axis  of  the  curve  ;  in  the  hyperbola  it  is  the  differ- 
ence of  these  two  lines  (F"N'  —  FN')  that  equals  the  major  axis  ; 
in  the  ellipse  the  eccentricity  is  less  than  unity  (zero  in  the  circle)  ;  in 
the  hyperbola  it  is  greater  than  unity;  in  the  parabola  exactly  unity. 

The  general  equation  of  a  conic  in  polar  co-ordinates,  applying  alike  to 
both  the  species,  is 

_          p 
1  -f-  e  cos  V3 


in  which  r  is  the  distance  Fn,  or  Fn',  e  is  the  fraction  —  f  Or  -  ,  the  angle  V 

-L   O          Jr  O 

is  the  angle  PFn,  PFn',  or  PFn",  and  p  is  the  line  FY,  FT,  or  FT',  called 
the  "semi-parameter."  The  word  "joarameter  "  means  the  cross  measure 
of  a  curve,  just  as  "diameter"  means  the  through  measure  of  a  curve.  If  e 
is  zero,  the  curve  is  a  circle,  and  r  —  p.  If  e  <  1,  the  curve  is  an  ellipse  ;  if 
e  >  1,  the  curve  is  an  hyperbola;  if  e  =  1,  it  is  a  parabola. 

A  generalized  form  of  Kepler's  third  law,  applying  to  hyperbolic  and 
parabolic  orbits  (which  have  no  periods)  as  well  as  elliptical,  jsjhjs  :  —  The 
Areal  velocities  of  bodies  revolving  around  the  sun  are  proportional  to  the 
square-roots  of  the  parameters  of  their  orbits. 


282  CENTRAL  FORCES. 

424.  Problem    of   Two    Bodies.  —  This   problem,    proposed   and 
solved  by  Newton,  is  the  following :  — 

Given  the  masses  of  two  spheres  and  their  positions  and  motions  at 
any  moment;  given,  also,  the  law  of  gravitation :  required  their  motion 
ever  afterwards,  and  the  data  necessary  to  compute  their  place  at  any 
future  time. 

The  mathematical  methods  by  which  the  problem  is  solved  require 
the  use  of  the  calculus,  and  must  be  sought  in  works  on  analytical 
mechanics  or  theoretical  astronomy.  Some  of  the  results,  however, 
are  simple  and  easily  stated. 

425.  (1)  In  the  first  place  the  motion  of  the  centre  of  gravity  of 
the  two  bodies  will  not  be  affected. by  their  mutual  attraction,  but  it 
will  move  on  uniformly  through  space,  as  if  the  bodies  were  united 
into  one  at  that  point,  and  their  motions  combined  under  the  same 
laws  which  hold  good  in  the  case  of  the  collision  of  inelastic  bodies. 

The  motion  of  this  centre  of  gravity  is  most  easily  worked  out  graphically 
as  follows :  First,  in  Fig.  141,  join  the  original  places  of  the  bodies  A  and 
B  by  a  straight  line,  and  mark  on  it  G,  the  place  of  the  centre  of  gravity ; 
then  take  the  positions  A'  and  B1  they  would  occupy  at  the  end  of  a  unit 
of  time  (if  they  did  not  attract  each  other),  and  mark  the  new  position  of 
the  centre  of  gravity  G1  on  the  line  joining  them.  The  line  GG'  connecting 


«* 


\ 

\B" 

FIG.  141.  —Motion  of  Bodies  relative  to  their  Centre  of  Gravity. 

the  two  positions  of  the  centre  of  gravity  will  show  the  direction  and  rapidity 
of  its  motion ;  with  reference  to  this  point  the  two  bodies  will  have  opposite 
motions  proportional  to  their  distances  from  it ;  that  is,  they  will  swing 
around  this  point  as  if  on  a  rod  pivoted  there,  and  will  either  both  move 
towards  it  along  the  rod,  or  from  it,  with  speeds  inversely  proportional 
to  their  masses.  These  relative  motions  with  respect  to  the  centre  of  gravity 
are  easily  found  by  drawing  through  G  a  line  parallel  to  A'Br,  and  meas- 
uring off  on  it  distances  GA"  and  GB"  respectively  equal. to  G'A'  and  G'B\ 
A  A"  and  BB"  will  then  be  the  two  motions  of  A  and  B  relative  to  their  centre 
of  gravity  G. 


EFFECT   OF  MUTUAL  ATTRACTION.  283 

426.  The  Effect  of  their  Mutual  Attraction. —This  will  cause 
them  to  describe  similar  conies  around  this  centre  of  gravity ;   the 
size  of  their  two  orbits  being  inversely  proportional  to  their  masses. 
The  form  of  the  orbits  and  dimensions  will  be  determined  by  the 
combined  mass  of  the  two  bodies,  and  by  their  velocities  with  respect 
to  the  common  centre  of  gravity  ^ 

427.  The  Orbit  of  the  Smaller  relative  to  the  Centre  of  the  Larger. 

—  It  is  convenient  (though  it  is  not  necessary)  to  drop  the  consid- 
eration of  the  centre  of  gravity  of  the  two  bodies,  and  to  consider 
the  motion  of  the  smaller  one  around  the  centre  of  the  larger  one. 
In  reference  to  that  point,  it  will  move  precisely  as  if  its  mass  had 
been  added  to  that  of  the  larger  body,  while  itself  had  become  a  mere 
particle.  This  relative  orbit  will  in  all  respects  be  like  the  actual  one 
around  the  centre  of  gravity,  only  magnified  in  the  proportion  of 
M  4-  m  to  M ;  i.e.,  if  ra  is  y1^  of  Jf,  the  actual  orbit  around  @  will 
be  magnified  by  |^  to  produce  the  relative  orbit  around  M. 

428.  The  Orbit  determined  by  Projection. — Suppose  that  in  the 
figure  (Fig.  142)  the  body  P  is  moving  in  the  direction  of  the  arrow, 


FIG.  142.  —Elliptical  Orbit  determined  by  Projection. 

and  is  attracted  by  $,  supposed  to  be  at  rest.  P  will  thenceforward 
move  in  a  conic,  either  in  an  ellipse  or  hyperbola,  according  to 
its  velocity,  as  we  shall  see  in  a  moment.  S  being  at  one  focus  of 
the  curve,  the  other  focus  will  be  somewhere  on  the  line  PN,  which 
makes  the  same  angle  with  PQ  that  r  (SP)  does  (since  it  is  a  prop- 
erty of  the  conies  that  a  tangent-line  at  any  point  of  the  curve  makes 
equal  angles  with  the  lines  drawn  from  the  two  foci  to  that  point) . 


284  CENTRAL   FORCES. 

If  we  can  find  the  place  of  the  second  focus  F9  or  the  length  of  the 
line  PF  in  the  figure,  the  curve  can  at  once  be  drawn. 

Now,  it  can  be  proved,  though  the  demonstration  lies  beyond  our 
scope,  that  <x,  the  semi-major  axis  of  the  conic,  is  determined  by 
the  equation 

72         /?  _  1 Y  (Equation  1) 

\r      aj 

in  which  r  is  the  distance  SP,  Fis  the  velocity,  and  /A  is  the  attracting 
mass  at  S  expressed  in  proper  units. 

(See  Watson's  "Theoretical  Astronomy,"  p.  49 ;  only  for  p  he  writes  fc2(l  +  m)). 
V,  r,  and  /*  being  given,  of  course  a  can  be  found :  we  get 

(Equation  2) 


r2/x-rF2 

Then  by  subtracting  r  from  2a  we  shall  get  r',  or  the  distance  PF, 
if  the  curve  is  an  ellipse.  If  it  is  a  hyperbola,  a  will  come  out  nega- 
tive ;  and  to  find  r1  we  must  take  r'  —  2a  +  r  and  measure  it  off  to 
F',  on  the  other  side  of  the  line  of  motion.  In  either  case,  however, 
we  easily  find  the  other  focus,  and  the  line  drawn  through  the  foci 
will  be  the  line  of  apsides ;  a  point  half-way  between  the  foci  will  be 
the  centre  of  the  curve,  and  any  line  drawn  through  this  centre  will  be 
a  diameter.  Having  the  two  foci  and  the  major  axis  2  a,  i.  e.,  AA\ 
the  curve  can  at  once  be  drawn. 

429.  Expression  for  a  in  Terms  of  the  "  Velocity  from  Infinity,"  or 
"Parabolic  Velocity."  —  The  expression  for  a  admits  of  a  more  con- 
venient and  very  interesting  form.  It  is  shown  in  analytical  mechan- 
ics that  if,  under  the  law  of  gravitation,  a  particle  falls  towards  an 
attracting  body  whose  mass  is  /x,  from  one  distance  s  to  another  dis- 
tance r,  its  velocity  is  given  by  the  simple  equation 

/I      i\  i 

w2  =  2p  I J .  (Equation  3) 

1  If  the  difference  between  s  and  r  is  called  h,  this  equation  becomes 


Now  if  h  is  very  small  as  compared  with  r,  this  gives 


to2 


-G* 


which  is  the  same  as  the  usual  expression  for  the  velocity  of  a  falling  body  at  the 
earth's  surface,  viz.,  T7"2  =  2<?ft,  2g  being  replaced  by  the  fraction     -. 


EXPRESSION   FOR    a.  285 

If  in  this  equation  s  be  made  infinite,  w  does  not  also  become 
infinite  (that  is,  a  body  falling  from  an  infinite  distance  towards  the 
sun  will  not  acquire  an  infinite  velocity  until  it  actually  reaches  the 
centre  of  the  sun,  and  r  becomes  zero)  ;  but  we  get  in  this  case 


This  special  value  of  w  is  usually  called  "  the  velocity  from  infinity 
for  the  distance  r"  or  the  "parabolic  velocity"  (for  a  reason  which 
will  appear  very  soon).  U  is  generally  used  as  its  symbol  ;  therefore 

and  //,  =  \r  U2.  (Equation  4) 

The  parabolic  velocity  due  to  the  sun's  attraction  at  any  point  is  therefore  in- 
versely proportional  to  the  square-root  of  the  distance  from  the  sun.  The  sun's 
mass  is  such  that  at  the  distance  unity  (the  mean  distance  of  the  earth  from 
the  sun)  it  is  equal  to  26.16  miles  or  42.  1O  kilometres.  At  the  sun's  sur- 
face it  is  383.04:  miles  or  616.40  kilometres,  and  at  the  distance  of  Neptune 


it  is  still  4.77  miles.     Again,  since  U  =  -V  —  >  U  varies  directly  as  the  square- 

root  of  the  mass  of  the  attracting  centre  of  force.  If  the  sun's  mass  were 
halved,  the  parabolic  velocity  due  to  its  attraction  would  everywhere  be 
reduced  in  the  ratio  of  Vi  to  1»  i-e">  as  0.7071  to  1.  If  the  mass  were 
doubled,  U=  would  be  increased  in  the  ratio  of  V^  to  1,  i.e.,  as  1.4142  to  1. 
The  square  of  the  parabolic  velocity  at  any  point  is  simply  twice  the  gravi- 
tation potential  due  to  the  sun's  attraction  at  that  point.  The  "potential  "  may 
be  defined  as  the  energy  which  would  be  acquired  by  a  mass  of  one  unit, 
in  falling  to  the  point  in  question  from  a  place  where  the  potential  (and 
attraction)  is  zero,  i.e.,  from  infinity.  Now  £  m  V2  is  the  general  expression 
for  the  kinetic  energy  of  a  mass,  m,  moving  with  velocity  V;  if  in  this  expres- 
sion we  make  m  =  1,  and  V=  U,  we  shall  have,  for  the  case  in  hand.  Energy 
=  £  U2,  which  therefore  equals  the  Potential  at  the  point. 

430.  Relation  between  the  Velocity  and  the  Species  of  Conic 
described.  In  equation  2  substitute  for  p  its  value,  %rU2  from 
equation  (4),  and  we  get 

(Equations). 


From  this  equation  it  is  clear  how  the  velocity  determines  whether 
the  orbit  will  be  an  ellipse  or  an  hyperbola.  If  V2  is  less  than  U2, 
the  denominator  of  the  fraction  will  be  positive,  a  will  also  be  posi- 
tive, and  the  curve  will  be  an  ellipse;  i.e.,  if  the  velocity  of  the  body 


286 


CENTRAL   FORCES. 


P,  at  the  distance  r  from  the  central  body  S,  be  less  than  the  velocity 
acquired  by  the  body  falling  from  infinity  to  that  point,  the  body 
will  move  around  S  permanently  in  an  ellipse. 

If,  on  the  other  hand,  F2  is  greater  than  U2,  the  denominator  will 
become  negative,  a  will  also  come  out  negative,  and  the  orbit  will  be 
an  hyperbola.  In  this  case  P,  after  once  moving  past  S  at  the  peri- 
helion point,  will  go  off  never  to  return ;  and  it  will  recede  towards 
a  different  region  of  space  from  that  out  of  which  it  came,  because 
the  two  legs  of  the  hyperbola  never  become  parallel.  There  will  in 
this  case  be  no  permanent  connection  between  the  two  bodies.  They 
simply  pass  each  other,  and  then  part  company  forever. 

If  F2  exactly  equals  U2,  the  denominator  of  the  fraction  becomes 
zero,  a  comes  out  infinite,  and  the  curve  is  a  parabola.  In  this  case, 
also,  the  body  will  never  return ;  but  it  will  recede  from  the  sun  ulti- 
mately towards  the  same  point  on  the  celestial  sphere  as  that  from 
which  it  appeared  to  come,  since  the  two  legs  of  the  parabola  tend  to 
parallelism.  Obviously,  if  a  body  were  thus  moving  in  a  parabola, 
the  slightest  increase  of  its  velocity  would  transform  the  orbit  into  an 
hyperbola,  and  the  least  diminution  into  an  ellipse;  the  bearing  of 
which  remark  will  become  evident  when  we  come  to  deal  with  comets. 


FIG.  143.  —  Confocal  Conies  described  under  Different  Velocities  of  Projection. 


GRAPHICAL   CONSTRUCTIONS.  287 


rf        U2      \ 
431.     Again,  since         a==2(us—  F2/ 


all  bodies  having  the  same  velocity  F,  at  the  same  distance  r  from  the 
centre  of  force,  will  have  major  axes  of  the  same  length  for  their  orbits, 
no  matter  what  may  be  the  direction  of  their  motion. 

They  will  have  the  same  period  also,  the  expression  for  the  period  being 

(Watson,  p.  46,  Equation  28.) 

"VA* 

But  observe  that  when  a  is  negative,  i.e.,  in  the  hyperbola,  the  value  of  t 
becomes  imaginary ;  there  is  no  periodicity  in  that  case. 

If,  therefore,  a  body  moving  around  the  sun  were  to  explode  at  any 
point,  all  of  its  particles  which  did  not  receive  a  velocity  greater  than 
the  "parabolic  velocity"  would  come  around  to  the  same  point  again, 
and  those  which  were  projected  with  equal  velocities  would  come 
around  and  meet  at  the  same  moment,  however  widely  different  their 
paths  might  be. 

432.  Fig.  143  represents  the  orbits  which  would  be  described  by  five 
bodies  projected  at  0  with  different  velocities  along  the  line  OF,  the  distance 
OS  or  r  being  taken  as  unity,  as  well  as  the  parabolic  velocity  U2.  The 
squares  of  the  velocities  are  assumed  as  given  below,  with  the  resulting 
values  of  a  and  /. 

Fx2  =  ^ ;  whence  a1  =  f ;  and  r^  =  J. 

This  places  the  empty  focus  at  Fr 
For  the  next  larger  ellipse 


In  the  same  way    F82  =  f ;  a8  =  2 ;  r8'  =  3. 

F42  =  1 ;  al  =  oo  ;  r4'  =  oo  .     (Parabola.) 
F52  =  2  ;  «5  =  -  * ;  r/  =  -  2.     (Hyperbola.) 

433.  Fig.  144  shows  how  three  bodies  projected  at  P  with  equal  velocities, 
but  in  different  directions,  indicated  by  the  arrows,  describe  three  different 
ellipses ;  all,  however,  having  the  same  period,  and  the  same  length  of  semi- 
major  axis ;  namely,  a  —  2  r;  F2  being  taken  equal  to  f  U2. 

For  a  fourth  body,  F2  is  taken  as  =  |  U2,  and  with  the  direction  of  motion 
perpendicular  to  r.  This  body  will  move  in  a  perfect  circle,  a  coming  out 
equal  to  r,  when  F2  =  ^U2.  In  order  to  have  circular  motion,  both  con- 
ditions must  be  fulfilled ;  namely,  F2  must  equal  ^  U2,  and  the  direction  of 
motion  must  be  perpendicular  to  the  radius  vector. 


288 


CENTRAL   FOKCES. 


FlG.  144.  —Ellipses  of  the  Same  Periodic  Time. 

These  conditions  are  of  course  fulfilled  very  nearly  in  the  case 
of  the  planets,  since  they  move  nearly  in  circles.  Observe  also  that 
if  the  mass  of  the  sun  were  somehow  to  be  suddenly  reduced  until 
the  corresponding  new  value  of  U2  were  less  than  V2,  the  planets5 
orbits  would  at  once  become  parabolas,  j£ 


434. 


Velocity  of  a  Planet  at  Any  Point  in  its  Orbit  —  If  A  A 

(Fig.  145)  be  the  major  axis 
of  a  planet's  orbit,  and  KK' 
the  diameter  of  a  circle  de- 
scribed around  S  with  A  A'  as 
radius,  then  the  velocity  of  a 
planet  at  any  point,  N,  on  its 
orbit  is  equal  to  that  which  it 
would  have  acquired  by  falling 
to  N  from  the  point  n  on  the 
circumference  of  the  circle. 
The  demonstration  is  not  dif- 
ficult and  may  be  found  in 
No.  1426  of  the  "Astronomi- 
sche  Nachrichten." 


FIG.  145.  —Theorem  of  Whewell  and  Van  der  Kolk. 


435.    Projectiles  near  the 

Earth.  —  A  good  illustration 
of  the  principles  stated  above  is  obtained  by  considering  the  motion  of 


INTENSITY   OF    SOLAR   ATTRACTION. 


289 


bodies  projected  horizontally  from  the  top  of  a  tower  near  the  earth's  surface, 
supposing  the  air  to  be  removed  so  there  will  be  no  resistance  to  the  motion. 

The  "parabolic  velocity"  due  to 
the  earth's  attraction  equals  6.94 
miles  per  second  at  the  earth's  sur- 
face ;  i.e.,  a  body  falling  from  the 
stars  to  the  surface  of  the  earth, 
drawn  by  the  earth's  attraction  only, 
would  have  acquired  this  velocity 
on  reaching  the  earth's  surface. 

First.  If  a  body  be  projected 
with  a  very  small  velocity,  it  would 
fall  nearly  straight  downwards.  If 
the  earth  were  concentrated  at  the 
point  in  its  centre  so  that  the  body 
should  not  strike  its  surface,  it  would 
move  in  a  very  long  narrow  ellipse  FIG.  146.  —  Projectiles  near  the  Earth, 
having  the  centre  of  the  earth  at  the 

further  focus,  and  would  return  to  the  original  point  after  an  interval  of 
29.9  minutes. 

Second.  With  a  greater  velocity  the  orbit  would  be  a  wider  ellipse  with  a 
longer  period,  C  being  still  at  the  remoter  focus. 

Third.  V  —  U V^,  or  about  4.9  miles  per  second.  In  this  case  the  orbit 
of  the  body  would  be  a  perfect  circle,  and  the  period  would  be  lh  24m.7. 
It  will  be  remembered  that  we  found  that  if  the  earth's  rotation  were  17 
times  as  rapid,  thus  completing  a  revolution  in  Ih24m.7,  the  centrifugal 
force  at  the  equator  would  become  equal  to  gravity  (Art.  154).  Also,  Art. 
413  (4),  this  same  time,  lh  24m.7,  was  found  from  Kepler's  third  law  as  the 
period  of  a  satellite  revolving  close  to  the  earth's  surface. 

Fourth.  V  —  U  —  6.94  miles.  In  this  case  the  projectile  would  go  oif  in 
a  parabola,  never  to  return. 

Fifth.  F>  6.94.  In  this  case,  also,  the  body  would  never  return,  but 
would  pass  off  in  an  hyperbola. 

At  the  surface  of  each  of  the  other  planets,  the  "parabolic 
velocity"  due  to  its  attraction  is  as  follows:  Mercury,  2.2  miles  per 
second  (probably,  but  very  uncertain);  Venus,  6.6;  Mars,  rh£r; 
Jupiter,  37;  Saturn,  22;  Uranus,  13;  Neptune,  14.  ^ 

In  the  case  of  the  sun  and  moon,  as  already  stated  (Arts.  429, 
272*),  the  parabolic  velocities  are  383  and  1.5  miles,  respectively. 


436.  Intensity  of  Solar  Attraction,  -  -  The  attraction  between 
the  sun  and  the  earth  from  some  points  of  view  looks  like  a  very 
feeble  action.  It  is  only  able,  as  has  been  before  stated  (Art.  278), 
to  bend  the  earth  out  of  a  rectilinear  course  to  the  extent  of  about 


290  EXERCISES. 

one-ninth  of  an  inch  in  a  second,  while  she  is  travelling  nearly 
nineteen  miles ;  and  yet  if  it  were  attempted  to  replace  by  bonds 
of  steel  the  invisible  gravitation  which  holds  the  earth  to  the  sun, 
we  should  find  the  surprising  result  that  it  would  be  necessary  to 
cover  the  whole  surface  of  the  earth  with  wires  as  large  as  telegraph 
wires,  and  only  about  half  an  inch  apart  from  each  other,  in  order 
to  get  a  metallic  connection  that  could  stand  the  strain.  This  liga- 
ment of  wires  would  be  stretched  almost  to  the  breaking  point.  The 
attraction  of  the  sun  for  the  earth  expressed  as  tons  of  force  (not 
tons  of  mass,  of  course)  is  3600000  millions  of  millions  of  tons 
(36  with  seventeen  ciphers)  ;  and  similar  stresses  act  through  the 
apparently  empty  space  in  all  directions  between  all  the  different 
pairs  of  bodies  in  the  universe. 


EXERCISES  ON  CHAPTER  XII. 

y 

1.   Given  a  comet  moving  in  an  ellipse  with  the  eccentricity  0.5.     Com- 
pare the  velocities,  both  linear  and  angular,  at  the  perihelion  and  aphelion. 
J  Lin.    Vel.  at  perihelion  is  three  times  that  at  aphelion. 
\  Aug.  Vel.        «         «         nine  " 

V  2.   At  what  point  in  the  orbit  is  the  actual  linear  velocity  equal  to  the 
mean  velocity?  An$    At  the  extremity  of  the  minor  axis. 

V3.   Is  the  angular  velocity  at  that  point  equal  to  the  mean  angular 
velocity  ;  and  if  not,  why  not  ? 

•\l  4.  What  would  be  the  periodic  time  of  a  small  body  revolving  in  a  circle 
around  the  sun  close  to  its  surface  ?     (Apply  Kepler's  harmonic  law.) 

Ans.   2  h.  47.4  min. 

^5.   What  would  be  its  velocity? 

Ans.   270.8  miles  a  sec. 

-^6.   If  the  earth  had  a  satellite  with  a  period  of  eight  months  what  would 
its  distance  be  ?  .  Qf 


i  **TT.  If  Jupiter  were  reduced  to  a  mere  particle  how  much  would  its 
period  be  lengthened  ?  (Consider  its  mass  to  be  y^^  of  the  sun's,  and  see 
Art.  417.) 


EXERCISES.  291 

Let  x  be  the  new  period ;  then 

1049 
a;2  :  Z2  — —  =  r3  :  r3  —  1  : 1,  since  r  is  not  changed.    Whence, 

=  t  (1  +  £  X  TTi¥¥  +  etc.)  =  t  (1  +  -ffg-s)  verv  nearly. 


4000  a 

But  f  =  4332.6  days,  and  x  -  t  =  -f      £  =  2.067  days.     Ans. 


8.  How  much  longer  would  the  earth's  period  be  if  it  were  a  mere 
Particle?  Ans.  Jnffyjfv  of  a  year,  or  47.8  sec. 

9.  If  the  sun's  mass  were  a  hundred  times  greater  what  would  be  the 
parabolic  velocity  at  the  earth's  distance  from  it?     (Art.  429.) 

Ans.    Ten  times  its  present  value,  i.e.,  261.6  miles  a  sec. 


LO.    If  the  sun's  mass  were  reduced  50  per  cent  what  would  be  the  para- 
bolic velocity  at  the  distance  of  the  earth  ? 

Ans.   18.5  miles  a  sec. 

^11.  If  the  sun's  mass  were  to-be  suddenly  reduced  by  50  per  cent  or 
more,  what  would  be  the  effect  upon  the  now  practically  circular  orbits  of 
the  planets?  (See  Art.  430.) 

Ans.    They  would  become  parabolas  or  hyperbolas,  and  the  planets 
would  desert  the  sun. 

12.  What  would  be  the  effect  upon  the  orbit  of  the  earth  if  the  sun's 
mass  were  suddenly  doubled  ? 

.4ns.   It  would  immediately  become  an  eccentric  ellipse,  with  its 
aphelion  near  the  point  where  the  earth  was  when  the  change  occurred. 

13.  Let  Vr  be  the  velocity  in  an  orbit  at  a  point  where  the  radius 
vector  is  r,  and  let  Ur  and  U2a  be  the  parabolic  velocities  at  distances  r 
and  2a  from  the  sun,  a  being  the  semi-major  axis  of  the  orbit.  Show 
that  F2r  =  Uzr  ±  U\a.  The  plus  sign  applies  if  the  orbit  is  an  hyperbola ; 
the  minus,  if  it  is  an  ellipse.  (See  Equations  1  and  4,  Arts.  428  and  429.) 


292  THE   PROBLEM   OF    THREE   BODIES. 


CHAPTER  XIII. 

THE     PEOBLEM     OF     THREE     BODIES.  --  DISTURBING     FORCES: 
LUNAR   PERTURBATIONS    AND   THE   TIDES. 

437.  THE  problem  of  two  bodies  is  completely  solved  ;  but  if, 
instead  of  two  spheres  attracting  each  other,  we  have  three  or  more, 
given  completely  in  respect  to  their  positions,  masses,  and  velocities, 
the  general  problem  of  finding  their  subsequent  motions  and  predict- 
ing their  positipns  at  any  future  date  transcends  the  present  power 
of  our  mathematics. 

This  problem  of  three  bodies  is  in  itself  just  as  determinate  and 
capable  of  solution  as  that  of  two.  Given  the  initial  data,  —  that 
is,  the  positions,  masses,  and  motions  of  the  three  bodies  at  a  given 
instant,  —  then  their  motions  for  all  the  future,  and  the  positions 
they  will  occupy  at  any  given  date,  are  absolutely  predetermined, 
provided  no  forces  act  upon  them  except  their  mutual  gravitational 
attractions.  The  difficulty  of  the  problem  lies  simply  in  the  inad- 
equacy of  our  present  mathematical  methods,  and  it  is  altogether 
probable  that  some  time  in  the  future  this  difficulty  will  be  overcome, 
though  at  present  there  is  no  immediate  prospect  of  success  ;  the 
problem  is  one  of  extreme  complexity. 

438.  But  while  the  general  problem  of  three  bodies  is  thus  intract- 
able, all  the  special  cases  of  it  which  arise  in  the  consideration  of 
the  moon's  motion  and  in  the  motions  of  the  planets  have  been 
solved  by  special  methods  of  approximation.     Newton  himself  led 
the  way;    and  the  strongest  proof  of  the  truth  of  his  theory  of 
gravitation  lies  in  the  fact  that  it  not  only  accounts  for  the  regular 
elliptic  motions  of  the  heavenly  bodies,  but  also  for  the  apparent 
irregularities  of  these  motions. 

439.  The  Disturbing  Force,  —  In  the  case  where  two  bodies  are 
revolving  around  their  common  centre  of  gravity,  and  the  third  body 
is  either  very  much  smaller  than  the  central  one,  or  very  remote,  the 


WHY  THE   SUN   DOES   NOT  TAKE  AWAY  THE  MOON.      293 

motion  of  the  two  will  be  but  slightly  modified  by  the  action  of  the 
third ;  and  in  such  a  case  the  small  differences  between  the  actual 
motion  and  the  motion  as  it  would  be  if  the  third  body  were  not 
present,  are  technically  called  "  disturbances  "  and  "  perturbations,"1 
and  the  force  which  produces  them  is  called  the  "  disturbing  force." 
This  disturbing  force  is  not  the  attraction  of  the  disturbing  body,  but 
only  a  component  of  that  attraction,  and  usually  only  a  small  fraction 
of  it. 

The  disturbing  force  of  the  attracting  body  depends  upon  the  differ- 
ence  of  its  attraction  upon  the  two  bodies  it  disturbs;  difference  either 
in  amount  or  in  direction,  or  in  both.  For  instance,  if  the  sun 
attracted  the  earth  and  moon  exactly  alike  (i.e.,  equally  and  along 
parallel  lines) ,  it  would  not  disturb  their  relative  motions  in  the  least, 
no  matter  how  powerful  its  attraction  might  be.  The  sun's  maxi- 
mum disturbing  force  on  the  moon,  as  we  shall  see,  is  only  about  one 
eighty-ninth  of  the  earth's  attraction  ;  and  yet  the  sun's  attraction  for 
the  moon  is  actually  much  greater  than  that  of  the  earth. 

Since  the  sun's  mass  is  330,000  times  that  of  the  earth,  and  its  distance 
just  about  389  times  that  of  the  moon  from  the  earth,  its  attraction  on  the 

330  000 
moon  equals  the  earth's  attraction  x  — • — =  2.18 ;  i.e.,  the  sun's  attraction 

OO«7 

on  the  moon  is  more  than  double  that  of  the  earth. 

440.    Why  the  Sun  does  not  take  the  Moon  away  from  the  Earth. 

—  If  at  the  time  of  new  moon,  when  the  moon  is  between  the  earth 
and  sun,  the  sun  attracts  the  moon  more  than  twice  as  much  as  the 
earth  does,  it  is  a  natural  question  why  the  sun  does  not  draw  the 
moon  away  entirely,  and  rob  us  of  our  satellite.  It  would  do  so  if 
it  were  the  case  of  a  "tug  of  war"  ;  that  is,  if  earth  and  sun  were 
fixed  in  space,  pulling  opposite  ways  upon  the  moon  between  them. 
But  it  is  not  so ;  neither  sun  nor  earth  has  any  foothold,  so  to  speak ; 
but  all  three  bodies  are  free  to  move,  like  chips  floating  on  water,  and  the 
student's  difficulty  in  understanding  the  action  of  disturbing  forces 
usually  lies  in  his  failure  to  appreciate  the  effect  of  this  freedom. 
The  sun  attracts  the  earth  almost  as  much  as  he  does  the  moon,  and 
both  earth  and  moon  fall  towards  him  freely ;  though  of  course  this 

1  The  student  will  bear  in  mind  that  these  terms  ("perturbations"  and  "dis- 
turbances") are  mere  figures  of  speech;  that  philosophically  the  purely  elliptical 
motion  of  two  mutually  attracting  bodies  alone  in  space  is  no  more  "regular" 
than  the  (at  present)  incomputable  motion  of  three  or  more  attracting  bodies. 


294 


THE  PROBLEM  OF   THREE  BODIES. 


falling  motion  towards  the  sun  is  continually  combined  with  whatever 
other  motion  the  earth  or  moon  possesses.  The  only  effective  dis- 
turbance is  produced  by  the  fact  that,  in  the  case  considered,  the  new 
moon,  being  nearer  the  sun  than  the  earth  is  by  about  ^?  part  of  the 
whole  distance,  falls  towards  the  sun  a  trifle  faster  than  the  earth, 
and  so  on  that  account  the  curvature  of  its  orbit  toward  the  earth  is, 
for  the  time  being,  diminished. 

At  the  half-moon  the  two  bodies  are  equally  attracted  towards  the 
sun,  but  on  converging  lines  ;  and  so  as  they  fall  towards  the  sun 
they  approach  each  other  slightly  ;  and  for  this  reason,  at  quadrature, 
the  moon's  orbit  is  a  little  more  curved  towards  the  earth  than  it 
would  be  otherwise. 

441.  Diagram  of  the  Disturbing  Force.  —  A  very  simple  diagram 
enables  us  to  find  graphically  the  disturbing  force  produced  by  a 
third  body. 

(What  follows  applies  verbatim  et  literatim  to  either  of  the  two  diagrams 
of  Fig.  147.) 


K 


K 

FIG.  147.  — Determination  of  the  Disturbing  Force  by  Graphical  Construction. 

Let  E  be  the  earth,  M  the  moon,  and  S  the  disturbing  body  (the 
sun  in  this  case)  ;  and  let  the  sun's  attraction  on  the  moon  be  repre- 
sented by  the  line  MS.  On  the  same  scale  the  attraction  of  the  sun 
on  the  earth  will  be  represented  by  the  line  EG,  G  being  a  point  so 
taken  that  EG :  MS  =  MS2 :  ES2 ;  that  is,  — 

The  sun's  attraction  on  the  earth  is  to  the  sun's  attraction  on  the  moon  as  the 
square  of  the  sun's  distance  from  the  moon  is  to  the  square  of  the  sun's  distance 
from  the  earth,  according  to  the  law  of  gravitation.  (M S  has  to  do  double 
duty  in  this  proportion :  in  the  first  ratio  it  represents  a  force  ;  in  the  second, 
a  distance.) 

From  this  proportion          EG  =  MS  X        * 


DIAGRAM  OF  THE  DISTURBING  FORCE.  295 

In  figure  (a)  the  moon  is  nearer  to  the  sun  than  the  earth  is,  and  so  EG 
comes  out  less  than  MS.  In  figure  (&)  the  reverse  is  the  case,  and  therefore 
in  this  case  EG  is  larger  than  MS. 

Now  if  the  force  represented  by  the  line  MS  were  parallel  and  equal 
to  that  represented  by  EG,  there  would  be  no  disturbance,  as  has  been 
said.  If,  then,  we  can  resolve  the  force  MS  into  two  components, 
one  of  which  is  equal  and  parallel  to  EG,  this  component  will  be  in- 
nocent and  harmless,  and  the  other  one  will  make  all 'the  disturbance. 

To  effect  this  resolution,  draw  through  M  the  line  MK  parallel 
and  equal  to  EG.  Join  KS,  and  draw  ML  parallel  and  equal  to  it. 
ML  is  then  the  disturbing  force  on  the  same  scale  as  MS;  i.e.,  the  line 
ML  shows  the  true  direction  of  the  disturbing  force,  and  in  amount 
the  disturbing  force  is  equal  to  the  sun's  attraction  for  the  moon  mul- 
tiplied "by  the  fraction  I )•  The  diagonal  of  the  parallelogram 

\MSJ 

MLS  IT  is  MS,  which  represents  the  resultant  of  the  two  forces  MK 
and  ML,  that  form  its  sides. 

For  the  sake  of  clearness  the  lines  which  represent  forces  in  the  figures 
are  indicated  by  herring-bone  markings. 

442.  At  first  it  seems  a  little  strange  that  in  figure  (6)  the  dis- 
turbing force  should  be  directed  away  from  the  sun;   but  a  little 
reflection  justifies  the  result.     If  E  and  M  were  connected  by  a  rod, 
and  the  E-eud  of  the  rod  were  pulled  towards  the  right  more  swiftly 
than  the  Jf-end,  it  is  easy  to  see  that  the  latter  would  be  relatively 
thrown  to  the  left,  as  the  figure  shows. 

443.  The  sun  is  the  only  body  that  sensibly  disturbs  the  moon. 
The  planets,  of  course,  act  upon  the  moon  to  disturb  it,  but  their 
mass  is  so  small  compared  with  that  of  the  sun,  and  their  distances 
so  great,  that  in  no  case  is  their  direct  action  sensible.     It  is  true, 
however,  that  some  of  the  lunar  perturbations  are  affected  by  the 
existence  of  one  or  two  of  the  planets.     While  they  cannot  disturb 
the  moon  directly,  they  do  so  indirectly:  they  disturb  the  earth  in 
her  orbit  sufficiently  to  make  the  sun's  action  different  from  what  it 
would  be  if  the  planets  did  not  exist,  and  in  this  way  make  them- 
selves felt.     There  are  also  a  few  small  disturbances  that  depend 
upon  the  fact  that  the  earth  is  not  a  perfect  sphere. 

444.  Since  the  distance  of  the  sun  is  nearly  four  hundred  times 
that   of  the  moon  from  the  earth,  and  the  moon's  orbit  is  very 
nearly   circular,   the   construction   of    the    disturbing    force   ML, 


296 


THE   PKOBLEM   OF   THREE   BODIES. 


Fig.  147,  admits  of  considerable  simplification.  It  is  only  Decessary 
to  drop  the  perpendicular  MP  upon  the  line  that  joins  the  earth  and 
the  sun,  and  take  the  point  L  upon  this  line,  so  that  EL  equals  three 
times  EP.  The  line  ML  so  determined  will  then  very  approximately 
(but  not  exactly)  be  the  true  disturbing  force. 


To  prove  this  relation,  let  MS,  in  Fig.  147,  be  Z>,  ES=  R,  ME  =  r,  and 
EP  =  p,  also  R'—D  +  p,  very  nearly,  p  being  negative  when  MS  >  ES.     EG 

was  taken  equal  to 


Developing  this  expression,  we  have 


Since  p  is  very  small  as  compared  with  Z>,  all  the  terms  except  the  first 
nearly  vanish  both  in  numerator  and  denominator,  and  we  have 


EL  =  =  3p  (very  nearly). 

445.  Resolution  of  the  Disturbing  Force  into  Components.  —  In 
discussing  the  effect  of  the  disturbing  force  it  is  more  convenient  to 
resolve  it  into  three  components  known  as  the  radial,  the  tangential, 
and  the  orthogonal.  The  first  of  these  acts  in  the  direction  of  the 

A 


FIG.  148.  — Radial  and  Tangential  Components  of  the  Disturbing  Force. 

radius  vector,  tending  to  draw  the  moon  either  towards  or  from  the 
earth.  The  second,  the  tangential,  operates  to  accelerate  or  retard 
the  moon's  orbital  velocity. 

Fig.  148  exhibits  these  two  components  at  different  points  of  the  moon's 
orbit. 


COMPONENTS    OF,  THE   DISTURBING   FORCE.  297 

*  3 


The  orthogonal  component  has  no  existence  in  cases  where  the 
disturbing  body  lies  in  the  plane  of  the  disturbed  orbit  ;  but  when- 
ever it  lies  outside  of  that  plane,  the  disturbing  force  ML  will  gen- 
erally also  lie  outside  of  the  orbit-plane,  and  will  have  a  component 
tending  to  draw  the  moving  body  out  of  the  plane  of  its  orbit.  The 
motion  of  the  moon's  node  and  the  changes  of  the  inclination  of  its 
orbit  are  due  to  this  component  of  the  sun's  disturbing  force,  which 
could  not  be  conveniently  represented  in  the  figure. 

446.  The  radial  force  in  the  case  of  the  moon's  orbit  is  a  maxi- 
mum at  syzygies  and  quadratures  ;  in  fact,  at  quadratures  the  whole 
disturbing  force  is  radial,  the  tangential  and  orthogonal  components 
both  vanishing.  At  syzygies  (new  moon  and  full  moon)  the  radial 
force  is  negative;  that  is,  it  draws  the  moon  from  the  earth,  dimin- 
ishing the  earth's  attraction  by  about  one  eighty-ninth  *  of  its  whole 
amount. 

At  quadrature  or  half-moon  the  radial  force  is  positive;  and  since 
L  then  falls  at  E,  it  is  represented  by  the  line  QE,  and  is  just  half 
what  it  is  at  syzygies  ;  that  is,  it  equals  about  one  one  hundred  and 
seventy  -eighth  of  the  earth's  attraction. 

It  becomes  zero  at  four  points  54°  44'  on  each  side  of  the  line  of 
syzygies. 


This  angle  is  found  from  the  condition  that  the  disturbing  force 
etc.,  in  Fig.  148,  must  be  perpendicular  to  the  radius  EMl  at  this  point, 
which   gives  us   EPl  :  PlMl  :  :  P.M,  :  P,L,.      But  PlL1  =  2EPl',    therefore 


447.  The  tangential  component  starts  at  zero  at  the  time  of  full 
moon,  rises  to  a  maximum  at  the  critical  angle  of  45°  (having  at 
that  point  a  value  of  yi-g-  of  the  earth's  attraction),  and  disappears 
again  at  quadratures.  During  the  first  and  third  quadrants  this 
force  is  negative;  that  is,  it  retards  the  moon's  motion  ;  in  the  second 
and  fourth  it  is  positive  and  accelerates  the  motion. 


i  At  syzygies  ML  =  NL0  =  2  x  EN  (Fig.  147) ;  but  EN  =        .      Therefore 

389 

o 

ML  — of  the  sun's  attraction  on  the  moon.     Now  the  sun's  attraction  is  2.18 

389 

o  Y  o  18        1 

times  the  earth's ;  hence  NL0=  the  earth's  attraction  multiplied  by     ^    ' —  =  — — . 

ooy         o9.2 


THE   PROBLEM   OF   THREE   BODIES. 


298 


448.  Lunar  Perturbations.  —  So  far  it  has  been  all  plain  sailing, 
for  nothing  beyond  elementary  mathematics  is  required  in  determin- 
ing the  disturbing  force  at  any  point  in  the   moon's   orbit;   but  to 
determine  what  will  be  the  effect  of  this  continually  varying  force  after 
the  lapse  of  a  given  time,  upon  the  moon's  place  in  the  sky  is  a  problem 
of  a  very  different  order,  and  far  beyond  our  scope.     The  reader  who 
wishes  to  follow  up  this  subject  must  take  up  the  more  extended 
works    upon  theoretical  astronomy   and  the  lunar  theory.      A  few 
points,  however,  may  be  noted  here. 

449.  In  the  first  place,  it  is  found  most  convenient  to  consider 
the  moon  as  never  deviating  from  an  elliptical  orbit,  but  to  consider 
the  orbit  itself  as  continually  changing  in  place  and  form,  writhing  and 
squirming,  so  to  speak,  under  the  disturbing  forces ;  just  as  if  the 
orbit  were  a  material  hoop  with  the  moon  strung  upon  it  like  a  bead 
and  unable  to  get  away  from  it,  although  she  can  be  set  forward  and 
backward  in  her  motion  upon  it. 

450.  In  the  next  place,  it  is  found  possible  to  represent  nearly  all 
the  perturbations  by  periodical  formula}  —  the  same  values  recurring 
over  and  over  again  indefinitely  at  regular  intervals.     This  is  because 
the  sun,  moon,  and  earth  keep  coming  back  into  the  same,  or  nearly 
the  same,  relative  positions,  and  this  leads  to  recurring  values  of  the 
disturbing  force  itself,  and  also  of  its  effects. 

451.  Third,  the  number  of  these  perturbations,  each  character- 
ized by  its  own  special  period,  is  very  large.    In  the  computation  of 
the  moon's  longitude  in  the  American  Ephemeris  about  seventy  dif- 
ferent inequalities  are  reckoned  in,  and  about  half  as  many  in  the 
computation  for  the  latitude.     Theoretically  the  number  is  infinite, 
but  only  a  certain  number  produce  effects  sensible  to  observation. 
It  is  of  no  use  to  compute  disturbances  that  do  not  displace  the  moon 
as  much  as  one-tenth  of  a  second  of  arc ;  i.e.,  about  500  feet  in  her 
orbit. 

452.  Fourth,  in  spite  of  all  that  has  been  done,  the  lunar  theory 
is  still  incomplete,  or  in  some  way  slightly  erroneous.     The  best 
tables  yet  made  begin   to   give  inaccurate  results  after  fifteen  or 
twenty  years,   and  require  correction.     The  almanac  place  of  the 
moon  at  present  is  not  unfrequently  "  out"  as  much  as  3"  or  4"  of 
arc  ;  i.e.,  about  three  or  four  miles.     Astronomers  are  continually  at 


LtTHAR   PERTURBATIONS.  299 

work  on  the  subject,  but  the  computations  by  our  present  methods 
are  exceedingly  tedious  and  liable  to  numerical  error. 

453.  The  principal  effects  of  the  sun's  disturbing  action  on  the 
moon  are  the  following  :  — 

First  :  Effect  on  Length  of  Month.  —  Since  the  radial  component 
of  the  disturbing  force  is  negative  more  than  half  the  way  round  (54° 
44'  on  each  side  of  the  line  of  syzygies)  and  is  twice  as  great  at 
syzygies  as  the  positive  component  is  at  quadrature,  the  net  result 
is  that,  taking  the  whole  month  through,  the  earth's  attraction  for 
the  moon  is  lessened  by  nearly  ^^  part.  The  effect  is  substantially 
the  same  that  would  follow  from  a  corresponding  diminution  of  the 
earth's  mass,  and  the  moon's  period  is  therefore  made  about  T^  part, 
or  nearly  an  hour,  longer  than  it  otherwise  would  be  at  its  present 


distance,  since  t  =  --~  •     (Art.  431.) 
Vf* 

454.  Second  :  The  Revolution  of  the  Line  of  Apsides.  —  This  is 
due  mainly  to  the  radial  component  of  the  disturbing  force,  though 
the  tangential  component  assists.    When  the  moon  comes  to  perigee 
or  apogee  at  the  time  of  new  or  full  moon  (i.e.,  during  those  months 
when  the  sun  is  at  or  near  the  line  of  apsides 

of   the   lunar   orbit),  the  diminution   of   the 

earth's  effective  attraction  for  the  moon  causes 

it  to  move  on  farther  than  it  would  otherwise 

do  before  turning  the  corner,  so  to  speak,  the 

consequence  being  that  the  line  of  apsides  ad- 

vances in  the  line  of  the  moon's  motion.   When 

perigee  or  apogee  is  passed  at  the  time  of 

quadrature,  the  line  of  apsides  is  also  disturbed 

and  made  to  regress.     But  at  quadrature  the 

radial  component  is  only  half  as  great  as  at 

syzygies,  and  the  net  result,  as  has  been  stated  before  (Art.  238),  is 

that  the  line  of  apsides  completes  a  direct  revolution  once  in  about 

nine  years  (8.855  years  —  Neison).    It  does  not  move  forward  steadily 

and  uniformly,  but  its  motion  is  made  up  of  alternate  advance  and 

regression.     Fig.  149  illustrates  this  motion  of  the  moon's  apsides. 

For  a  fuller  discussion  of  the  subject,  see  Herschel's  "  Outlines  of  Astron- 
omy," Sections  677-689  ;  or  Airy's  «  Gravitation,"  pp.  89-100. 

455.  Third  :   The  Regression  of  the  Nodes.  —  The  orthogonal  com- 
ponent generally  (not  always)  tends  to  draw  the  moon  towards  the  plane 


300  THE   PROBLEM   OF   THREE   BODIES. 

of  the  ecliptic.  Whenever  this  is  the  case  at  the  time  when  the  moon 
is  passing  a  node,  the  effect  (as  is  easily  seen  from  Fig.  150)  of  such 
a  force  P\0^  acting  upon  the  moon  at  P1?  is  to  shift  the  node  back- 
ward from  NI  to  N2,  the  moon  taking  the  new  path  PlblN2.  As  the 
moon  is  approaching  the  node,  the  inclination  of  its  orbit  is  also  in- 
creased ;  but  as  the  moon  leaves  the  node,  it  is  again  diminished,  the 
path  N2P2  being  bent  at  P2  back  to  P2b.2,  parallel  to  P^ :  so  that 
while  by  both  operations  the  node  is  made  to  recede  from  N!  to  N8, 
the  inclination  suffers  very  little  change,  if  the  orthogonal  component 
remains  the  same  on  both  sides  of  the  node. 

Since  the  orthogonal  component  vanishes  twice  a  year,  —  when  the 
sun  is  at  the  nodes  of  the  moon's  orbit,  —  and  also  twice  a  month, 
— when  she  is  in  quadrature,  —  the  rate  at  which  the  nodes  regress 


E 


c 


iV, 


Fie.  150.  — Regression  of  the  Nodes  of  the  Moon's  Orbit. 

is  extremely  variable.  In  the  long  run  it  makes  its  backward  revo- 
lution once  in  about  nineteen  years  (Arts.  249  and  391).  [18.5997 
years.  —  Neison.~\ 

See  Herschel's  "  Outlines  of  Astronomy,"  section  638  seqq. 

456.  Fourth:  The  Evection. — This  is  an  irregularity  which  at  the 
maximum  puts  the  moon  forward  or  backward  about  H°(l°  16'27".01 — 
Neison),  and  has  for  its  period  the  time  which  is  occupied  by  the 
sun  in  passing  from  the  line  of  apsides  of  the  moon's  orbit  to  the 
same  line  again  ;  i.e.,  about  a  year  and  an  eighth.  This  is  the  largest 
of  the  moon's  perturbations,  and  was  earliest  discovered,  having  been 
detected  by  Hipparchus  about  150  years  B.C.,  and  afterwards  more 
fully  worked  out  by  Ptolemy,  though  of  course  without  any  under- 
standing of  its  cause.  It  was  the  only  lunar  perturbation  known  to 
the  ancients.  It  depends  upon  the  alternate  increase  and  decrease  of 
the  eccentricity  of  the  moon's  orbit,  which  is  always  a  maximum  when 


EVECTION.  301 

the  sun  is  passing  the  moon's  line  of  apsides,  and  a  minimum  when 
the  sun  is  at  right  angles  to  it. 

This  inequality  may  affect  the  time  of  an  eclipse  by  nearly  six 
hours,  making  it  anywhere  from  three  hours  early  to  three  hours  late, 
as  compared  with  the  time  at  which  it  would  otherwise  occur ;  it  was 
this  circumstance  which  called  the  attention  of  Hipparchus  to  it. 

See  Herschel's  "  Outlines  of  Astronomy,"  sections  748  seqq. 

457.  Fifth:  The  Variation.  —  This  is  an  inequality  due  mainly  to 
the  tangential  component  of  the  disturbing  force.     It  has  a  period  of 
one  month,  and  a  maximum  amount  of   39'  30". 70,  attained  when 
the  moon  is  half-way  between  the  syzygies  and  quadratures,  at  the 
so-called  u  octants."     At  the  first  and  third  octants  the  moon  is  39^-' 
ahead  of  her  mean  place  (about  an  hour  and  twenty  minutes)  ;  at  the 
second  and  fourth  she  is  as  much  behind.     This  inequality  was  de- 
tected by  Tycho  Brahe,  though  there  is  some  reason  for  believing 
that  it  had  been  previously  discovered  by  the  Arabian  astronomer, 
Aboul  Wefa,  five  centuries  earlier.     This  inequality  does  not  affect 
the  time  of  an  eclipse,  being  zero  both  at  the  syzygies  and  quadra- 
tures, and  therefore  was  not  detected  by  the  Greek  astronomers. 

See  Herschel's  "  Outlines  of  Astronomy,"  sections  705  seqq. 

458.  Sixth:    The  Annual  Equation.  —  The   one   remaining  ine- 
quality which  affects  the  moon's  place  by  an  amount  visible  to  the 
naked  eye,  is  the  so-called  "annual  equation."     When  the  earth  is 
nearer  the  sun  than  its  mean  distance,  the  sun's  disturbing  force  is, 
of  course,  greater  than  the  mean,  and  the  month   is   lengthened  a 
little;  during  that  half  of  the  year,  therefore,  the  moon  keeps  falling 
behindhand ;  and  vice  versa  during  the  half  when  the  sun's  distance 
exceeds  the  mean.      The   maximum  amount   of    this*  inequality  is 
11 '  9 ".00,  and  its  period  one  anomalistic  year. 

See  Herschel's  "  Outlines  of  Astronomy,"  sections  738  seqq. 

There  remains  one  lunar  irregularity  among  the  multitude  of  lesser 
ones,  which  is  of  great  interest  theoretically,  and  is  still  a  bone  of 
contention  among  mathematical  astronomers  ;  namely,  — 

459.  Seventh :  The  Secular  Acceleration  of  the  Moon's  Mean  Mo- 
tion. —  It  was  found  by  Halley,  early  in  the  last  century,  by  a  com- 
parison  of    ancient  with    modern   eclipses,   that  the  month  is  now 


302  THE  PROBLEM   OF   THREE   BODIES. 

certainly  shorter  than  it  was  in  the  days  of  Ptolemy,  and  that  the 
shortening  has  been  progressive,  apparently  going  on  continuously, 
—  in  scecula  sceculorum,  —  whence  the  name.  In  100  years  the 
moon,  according  to  the  results  of  Laplace,  gets  in  advance  of  its 
mean  place  about  10",  and  the  advance  increases  with  the  square  of 
the  time,  so  that  in  a  thousand  years  it  would  gain  nearly  1000", 
and  in  2000  }*ears  4000",  or  more  than  a  degree.  The  moon  at 
present  is  supposed  to  be  just  about  a  degree  in  advance  of  the  posi- 
tion it  would  have  held  if  it  had  kept  on  since  the  Christian  era 
with  precisely  the  rate  of  motion  it  then  had.  If  this  acceleration 
were  to  continue  indefinitely,  the  ultimate  result  would  be  that  the 
moon  would  fall  upon  the  earth,  as  the  quickened  motion  corresponds 
to  a  shortened  distance. 

460.  It   was  nearly  100  years   after  Halley's   discovery  before 
Laplace  found  its  explanation  in  the  decreasing  eccentricity  of  the 
earth's  orbit.     Under  the  action  of  the  other  planets  this  orbit  is  now 
growing  more  nearly  circular,  without,  however,  changing  the  length 
of  its  major  axis.    Thus  its  area  becomes  larger,  and  the  earth's  average 
distance  from  the  sun  becomes  greater  (although  the  mean  distance, 
technically  so-called,  does  not  change,  the  "mean  distance"  being 
simply  half  the  major  axis) .     As  a  result  of  this  rounding  up  of  the 
earth's  orbit,  the  average  disturbing  force  of  the  sun  is  therefore  dimin- 
ished, and  this  diminution  allows  the  month  to  come  nearer  the  length 
it  would  have  if  there  were  no  sun  to  disturb  the  motion  ;  that  is  to 
say,  the  month  keeps  shortening  little  by  little,  and  it  will  continue 
to  do  so  until  the  eccentricity  of  the  earth's  orbit  begins  to  increase 
again,  some  25,000  years  hence. 

461.  But  the   theoretical  amount  of  this  acceleration,  about  6"  in  a 
century,  does  not  agree  with  the  value  obtained  by  comparing  the  most 
ancient  and  modern  eclipses,  which  is  about  12" ;  and  this  value,  again,  does 
not  agree  with  the  one  derived  by  comparing  modern  observations  of  the 
moon  with  those  made  by  the  Arabians  about  a  thousand  years  ago,  which, 
according  to  recent   investigations   by   Professor   Newcomb,   indicate    an 
acceleration  of  only  about  8fr. 

So  long  as  the  actual  acceleration  was  considered  to  be  12",  it  was  gener- 
ally supposed  that  the  discrepancy  between  the  theoretical  and  observed 
result  is  due  to  a  retardation  of  the  earth's  rotation  by  the  friction  of  the 
tides,  and  a  consequent  lengthening  of  the  day.  Evidently  if  the  day  and 
the  seconds  become  a  little  longer,  there  will  be  fewer  of  them  in  each 
month  or  year,  and  the  apparent  effect  of  such  a  change  would  be  to  shorten 
all  really  constant  astronomical  periods  by  one  and  the  same  percentage. 


THE   TIDES.  303 

As  matters  stand  to-day  it  is  hardly  possible  to  assert  with  confidence 
that  there  is  any  real  discrepancy  to  be  accounted  for  between  the  theoret- 
ical and  observed  values,  the  latter  being  considerably  uncertain.  In  New- 
comb's  "  Popular  Astronomy  "  (pp.  96-102)  there  will  be  found  an  interesting 
and  trustworthy  discussion  of  the  subject. 

Questions  like  this,  and  those  relating  to  the  remaining  discrepancies 
between  the  lunar  tables  and  the  observed  places  of  our  satellite,  lie  on  the 
very  frontiers  of  mathematical  astronomy,  and  can  be  dealt  with  only  by 
the  ablest  and  most  skilful  analysts. 

THE   TIDES. 

462.  Just  as  the   disturbing   force   due  to   the   sun's   attraction 
affects   the   motions   of   the   moon    in    her   orbit,  so  the  disturbing 
forces  due  to  the  attractions  of  the  moon  and  sun  acting  upon  the 
fluids  of  the  earth's  surface  produce  the  tides.     These  consist  of  the 
regular  rise  and  fall  of  the  water  of  the  ocean  usually  twice  a  day, 
the  average  interval  between  the  corresponding  high  waters  of  suc- 
cessive days  being  24h  51m,  which  is   precisely  the   same  as  the 
average  interval  between  two  successive  passages  of  the  moon  across 
the  meridian.     This  coincidence,  maintained  indefinitely,  of  itself 
makes  it  certain  that  there  must  be  some  causal  connection  between 
the  moon  and  the  tides. 

463.  Definitions.  — When  the  water  is  rising,  it  is  " flood"  tide  ; 
when  falling,  it  is  "  ebb."     It  is  "  high  water"  at  the  moment  when 
the  tide  is  highest,  and  "low  water"  when  it  is  lowest.     "  Spring 
tides"  are  the  highest  tides  of  the  month  (which  occur     near  the 
times  of  new  and  full  moon),  while  "neap  tides"  are  the  smallest, 
which  occur  when  the  moon  is  in  quadrature.     The  relative  heights  of 
the   spring   and   neap   tides   are*  about    as    7    to    4.     At  the   time 
of  spring  tides  the  interval  between  the  corresponding  tides  of  suc- 
cessive days  is  less  than  the  average,   being   only  about  24h   38ln, 
and  then  the  tides  are  said  to  "prime."     At  neap  tides  the  interval 
is  25h  6m,  which  is  greater  than  the  mean,  and  the  tides  "  lag." 

-TJie  "establishment"  of  a  port  is  the  mean  interval  between  the 
time  of  high  water  at  that  port  and  the  next  preceding  passage  of 
the  moon  across  the  meridian.  At  New  York,  for  instance,  this 
"  establishment"  is  8h  13m,  although  the  actual  interval  varies  about 
22  minutes  on  each  side  of  the  mean  at  different  times  of  the  month. 

That  the  moon  is  largely  responsible  for  the  tides  is  also  shown  by 
the  fact  that  the  tides,  at  the  time  when  the  moon- is  in  perigee,  are 
nearly  twenty  per  cent  higher  than  those  which  occur  when  she  is  in 


304  THE   TIDES. 

apogee.  The  highest  tides  of  all  happen  when  a  new  or  full  moon 
occurs  at  the  time  the  moon  is  in  perigee,  especially  if  this  occurs 
about  January  1st,  when  the  earth  is  nearest  to  the  sun.  Since,  as 
we  shall  see,  the  "  tide-raising  "  force  varies  inversely  as  the  cube  of 
the  distance,  slight  variations  in  the  distance  of  the  moon  and  sun 
from  the  earth  make  much  greater  variations  in  the  height  of  the  tide 
—  greater  nearly  in  the  ratio  of  3  to  1. 

464.  The  Tide-Raising  Force. — This  is  the  difference  between  the 
attractions  of  the  sun  and  moon  (mainly  the  latter)  on  the  main 
body  of  the  earth,  and  the  attractions  of  the  same  bodies  on  parti- 
cles at  different  parts  of  the  earth's  surface.  The  tide-raising  force 
is  but  a  very  small  part  of  the  whole  attraction. 

The  amount  of  this  disturbing  force  for  a  particle  at  any  point 
on  the  earth's  surface  can  be  found  approximately  by  the  same  geo- 


E 

FIG.  151.— The  Moon's  Tide-Raising  Force  on  the  Earth. 

metrical  construction  which  was  used  for  the  lunar  theory  (Art. 
441).  Draw  a  line  from  the  moon  through  the  centre  of  the  earth. 
At  the  points  A  and  B,  Fig.  151,  where  the  moon  is  directly  over 
head  or  under  foot,  the  tide-raising  force  is  directly  opposed  to  grav- 
ity, and  equals  nearly  fa  of  the  moon's  whole  attraction,  since  the  line 
Aa  represents  the  disturbing  force  on  the  same  scale  as  the  line  from 
A  to  the  moon  represents  the  moon's  attraction,  and  this  line,  AM, 
is  about  sixty  times  the  earth's  radius,  while  Aa  is  just  double  it, 
because  Ca  has  to  be  taken  equal  to  3  x  CA  (Art.  444). 

Since  the  moon's  mass  is  only  about  fa  of  the  earth's,  and  its  dis- 
tance is  sixty  radii  of  the  earth,  this  lifting  force  under  the  moon, 
expressed  as  a  fraction  of  the  earths  gravity,  equals 

sV  X  s*o  x  3sVi7 =  -5-5* V-GW  5 

i.e.,  a  body  weighing  four  thousand  tons  loses  about  one  pound  of  its 
weight  when  the  moon  is  over  head  or  under  foot. 

At  D  and  E,  anywhere  on  the  circle  of  the  earth's  surface  which 
is  90°  from  A  and  B,  the  moon's  disturbing  force  increases  the 


i 


THE    TIDE-RAISING    FORCE.  305 

weight  of  a  body  by  just  half  this  amount,  the  disturbing  force  being 
measured  by  the  lines  DC  and  EC.  At  a  point  F,  situated  anywhere 
on  a  circle  drawn  around  either  A  or  B  with  a  radius  of  54°  44',  the 
weight  of  a  body  is  neither  increased  nor  decreased,  but  it  is  urged 
towards  A  or  B  with  a  horizontal  force  expressed  by  the  line  Ff, 
which  force  is  equal  to  about  T^-o  (bnoo  °^  ^s  weignt- 

The  tidal  forces  at  G  and  ^Tare  expressed  by  the  lines  Gg  and  Hh, 
each  resolvable  into  vertical  and  horizontal  components. 

465,  The  same  result  for  the  lifting-force  directly  under  the  moon  may 
be  obtained  more  exactly  as  follows.  The  distance  from  the  moon  to  the 
centre  of  the  earth  is  sixty  times  the  earth's  radius,  and  therefore  the 
distance  from  the  moon  to  the  points  A  and  B  respectively  will  be  59  and 
61.  The  moon's  attraction  at  A,  C,  and  B,  expressed  as  fractions  of  the 
earth's  gravity,  will  be  as  follows  :  — 

Attraction  of  moon  on  particle  at  A  =  g  x  -^L  —  0.0000035910  X  g. 

59 

Attraction  of  moon  on  particle  at  C  =  g  x  A  =  0.0000034723  X  g. 


=     x  -£1L  — 
Hence,  A  -  C  =  0.0000001187  g  = 


Attraction  of  moon  on  particle  at  B  =  g  x  -£1L  —  0.0000033593  X  g. 


C-£  =  0.0000001130,  = 

This  is  more  correct  than  the  preceding,  which  is  based  on  an  approxima- 
tion that  considers  the  moon's  distance  as  very  large  compared  with  the 
earth's  radius,  while  it  is  really  only  sixty  times  as  great,  and  sixty  is  hardly 
a  "  very  large  "  number  in  such  a  case. 

Attempts  have  been  made  to  observe  directly  the  variations  in  the  force  of 
gravity  produced  by  the  moon's  action,  but  they  are  too  small  to  be  detected 
with  certainty  by  any  experimental  method  yet  contrived.  Both  Darwin 
and  Zollner  found  that  other  causes  which  they  could  not  get  rid  of  produced 
disturbances  more  than  sufficient  to  mask  the  whole  action  of  the  moon.  ,- 

466.  It  is  worth  while  to  note  in  this  connection  that  the  maximum 
lifting-force  due  to  the  attraction  of  a  distant  body  varies  inversely  as  the 
cube  of  its  distance,  as  is  easily  shown,  thus  :  —  calling  D  the  distance  of  the 
disturbing  body  from  the  earth's  centre,  and  r  the  earth's  radius,  we  have 

Attraction  at  A  =  -  —  -  ;       attraction  at  C  =  —• 


.  ,  (         1  1  )        ,  .  ( 

Tlde-ra1S1ng  force  at  A  =  M  j  ^—  _  -  -  j.  =  M  j 


2Dr  — 


when  r  is  a  small  fraction  of  D.  [/ 


306  THE  TIDES. 

467.  It  is  very  apt  to  puzzle  the  student  that  the  moon's  action 
should  be  a  lifting  force  at  B  as  well  as  at  A  (Fig.  151).     He  is 
likely  to  think  of  the  earth  as  fixed,  and  the  moon  also  fixed  and 
attracting  the  water  upon  the  earth,  in  which  case,  of  course,  the 
moon's  attraction,  while  it  would  decrease  gravity  at  A,  would  increase 
it  at  B. 

The  two  bodies  are  not  fixed, 
however.  Let  him  think  of  the  three 
particles  at  J.,  (7,  and  B,  Fig.  152, 
as  unconnected  with  each  other,  and 
falling  freely  towards  the  moon ; 
then  it  is  obvious  that  they  would 
separate ;  A  would  fall  faster  than 
Fie.  i52.-The  statical  Theory  of  the  Tides.  C,  and  C  than  B.  Now  imagine 

them  connected  by  an  elastic  cord. 

It  is  obvious  that  they  will  still  draw  apart  until  the  tension  of  the 
cord  prevents  any  further  separation.  Its  tension  will  then  measure 
the  "  lifting  force"  of  the  moon  which  tends  to  draw  both  the  par- 
ticles A  and  B  away  from  G. 

468.  The  Sun's  Action.  —This  is  precisely  like  that  of  the  moon, 
except  that  the  sun's  distance,  instead  of  being  only  sixty  times  the 
earth's  radius,  is  nearly  23,500  times  that  quantity.     Since  the  tide- 
raising  power  varies  as  the  cube  of  the  distance  inversely,  while  the 
attracting  force  varies  only  with  the  inverse  square,  it  turns  out  that 
although  the  sun's  attraction   on  the   earth  is  nearly  200  times  as 
great  as  that  of  the  moon,  its  tide-raising  power  is  only  about  two- 
fifths  as  much.     When  the  sun  is  over  head  or  under  foot,  his  dis- 
turbing force  diminishes  gravity  by  about  ia  6oTnnnr* 

469.  Statical  Theory  of  the  Tides. — If  the  earth  were  wholly 
composed  of  water,  and  if  it  kept  always  the  same  face  towards  the 
moon  (as  the  moon  does  towards  the  earth) ,  so  that  every  particle  on 
the  earth's  surface  was  always  subjected  to  the  same  disturbing  force 
from  the  moon,  then,  leaving  out  of  account  the  sun's  action,  a  per- 
manent tide  would  be  raised  upon  the  earth,  distorting  it  into  a 
lemon-shaped  form  with  the  point  towards  the  moon.     It  would  be 
permanently  high  water  at  the  points  A  and  B  (Fig.  152)  directly 
under  the  moon,  and  low  water  all  around  the  earth  on  the  circle  90° 
from  these  points,  as  at  D  and  E.     The  difference  of  the  level  of 
the  water  at  A  and  D  would  in  this  case  be  about  two  feet. 


THE   PRIMING  AND   LAGGING   OF   THE  TIDES.  307 

The  sun's  action  would  produce  a  similar  tide  superposed  upon 
the  lunar  tide  and  having  about  two-fifths  of  the  same  elevation.  If 
the  two  tide  summits  should  coincide,  the  resulting  elevation  of  the 
high  water  would  be  the  sum  of  the  two  separate  tides.  If  the  sun 
were  90°  from  the  moon,  the  waves  would  be  in  opposition,  and  the 
height  of  the  tide  would  be  decreased,  the  solar  tide  partly  filling  up 
the  depression  at  the  low  water  due  to  the  moon's  action. 

Suppose  now  the  earth  to  be  put  in  rotation.  It  is  easy  to  see 
that  these  tidal  waves  would  tend  to  move  over  the  earth's  surface, 
following  the  moon  and  sun  at  a  certain  angle  dependent  on  the 
inertia  of  the  water,  and  with  a  westward  velocity  precisely  equal  to 
that  of  the  earth's  eastward  rotation,  —  about  a  thousand  miles  an 
hour  at  the  equator.  But  it  is  also  evident  that  on  account  of  the 
varying  depth  of  the  ocean,  and  the  irregular  form  of  the  shores,  the 
tides  could  not  maintain  this  motion,  and  that  the  actual  result  must 
become  exceedingly  complicated.  In  fact,  the  statical  theory  be- 
comes utterly  unsatisfactory  in  regard  to  what  actually  takes  place, 
and  it  is  necessary  to  depend  almost  entirely  upon  the  results  of 
observation,  using  the  theory  merely  as  a  guide  in  the  discussion  of 
the  observations. 

Yet  while  this  statical  theory  of  the  tides  worked  out  by  Newton  is 
certainly  inadequate,  and  in  some  respects  incorrect,  it  easily  furnishes  the 
explanation  of  some  of  the  most  prominent  of  the  peculiarities  of  the 


470.  The  Priming  and  Lagging  of  the  Tides.  —  About  the  time 
of  new  and  full  moon,  as  has  been  stated  before  (Art.  463),  the  interval 
between  the  corresponding  tides  of  successive  days  is  about  thirteen  minutes 
less  than  the  average  of  24h  51m,  while  a  week  later  it  is  about  as  much 
longer.  The  reason  is  found  in  the  combination  of  the  solar  and  lunar  tides. 

L  L    X 


•i 

FIG.  153.  FIG.  154. 

Priming  and  Lagging  of  the  Tide. 


On  the  days  of  new  and  full  moon  the  two  tides  coincide,  and  the  tide 
wave  has  its  crest  directly  under  the  moon,  or  rather  at  the  normal  distance 
behind  the  moon  which  corresponds  to  the  "  establishment "  of  the  port  of 
observation. 


308 


THE  TIDES. 


At  quadrature  the  crest  of  the  solar  tide  will  be  just  90°  from  the  crest  of 
the  lunar  wave,  but  it  will  leave  the  summit  of  the  combined  wave  just  where 
It  would  be  if  there  were  no  solar  wave  at  all :  evidently  there  is  no  possible 
reason  why  the  smaller  wave  at  S  and  S1  should  displace  the  crest  of  the  wave 
at  L  (Fig.  153)  towards  the  right  that  would  not  also  require  its  displacement 
towards  the  left ;  it  will  therefore  simply  lower  the  wave  at  L  without  dis- 
placing it  one  way  or  the  other.  But  when  the  solar  tide  wave  SSf  (Fig. 
154)  has  its  crest  at  S1  and  £/,  45°  from  L  and  L',  as  it  will  do  about  three 
days  after  new  or  full  moon,  then  its  combination  with  the  lunar  wave  will 
make  the  crest  of  the  combined  wave  take  position  at  a  point  X  between  the 
two  crests,  and  about  half  an  hour  of  time  ahead  (west)  of  the  lunar  tide; 
so  that  at  that  time  of  the  month  high  water  will  occur  about  half  an  hour 
earlier  than  if  there  were  no  solar  tide  (since  the  tide  waves  travel  westward). 
And  this  half-hour  has  to  be  gained  by  diminishing  the  interval  between 
the  successive  tides  for  the  three  preceding  days.  Similar  reasoning  shows 
that  when  the  solar  tide  crest  falls  at  S2  and  £/,  the  combined  tide  wave  will 
be  east  of  the  lunar  wave,  and  come  later  into  port. 

471.  Effect  of  the  Moon's  Declination  and  Diurnal  Inequality.  — 

In  high  latitudes  on  the  Pacific  Ocean,  twice  a  month,  when  the  moon  is 
farthest  north  or  south  of  the  celestial  equator,  the  two  tides  of  the  day  are 

very  different  in  magnitude.  When  the 
moon's  declination  is  zero,  there  is  no 
such  difference:  nor  is  there  ever  any 
difference  at  ports  which  are  near  the 
earth's  equator. 

Fig.  155  makes  it  clear  why  it  should 
be  so.  When  the  moon's  declination  is 
zero,  things  are  as  in  Fig.  152  (Art. 
469),  and  the  two  tides  of  the  same 
day  are  sensibly  equal  at  ports  in  all 
latitudes.  When  the  moon  is  at  her 
greatest  northern  declination,  say  28°, 

the  two  tide  summits  will  be  at  A  and  A'  in  Fig.  155;  the  tide  which 
occurs  at  B  when  the  moon  is  overhead  will  be  great,  while  the  tide  in  the 
corresponding  southern  latitude  at  B1  will  be  small.  The  tides  which 
occur  twelve  hours  later  will  be  small  at  the  northern  station,  then  situated 
at  C,  and  large  at  the  southern  station,  then  at  C'.  For  a  port  on  the 
equator  at  E  or  Q  there  will  be  no  such  difference.  In  the  Atlantic  Ocean 
the  difference  is  hardly  noticeable,  because,  as  we  shall  see  very  soon,  the 
tides  in  that  ocean  are  mainly  (not  entirely)  due  to  tide  waves  propagated 
into  it  from  the  Pacific  and  Indian  Oceans  around  the  Cape  of  Good  Hope. 

472.  The  Wave  Theory  of  the  Tides.  —  If  the  earth  were  entirely 
covered  with  deep  water,  except  a  few  little  islands  projecting  here 
and  there  to  serve  for  observing  stations,  the  tide  waves  would  run 


FIG.  155.  —The  Diurnal  Inequality. 


FREE  AND   FORCED   OSCILLATIONS.  309 

around  the  globe  regularly.  According  to  Darwin  the  tide-crests, 
if  the  depth  exceeded  14  miles,  would  keep  exactly  under  the  moon 
(considering  only  the  lunar  tide).  If  the  depth  were  somewhat  less, 
the  tide-crests  on  the  equator  would  follow  the  moon  at  an  angle  of 
90°,  while  those  near  the  poles  would  maintain  their  old  relation,  and 
at  some  intermediate  latitude  there  would  be  a  tideless  belt  of  con- 
flicting currents.  In  the  actual  ocean,  comparatively  shallow  and 
of  varying  depth,  the  case  becomes  hopelessly  complicated.  The 
continents  of  North  and  South  America,  with  the  southern  antarctic 
continent,  make  a  barrier  almost  complete  from  pole  to  pole,  leaving 
only  a  narrow  passage  at  Cape  Horn ;  and  the  varying  depth  of  the 
water  and  the  irregular  contours  of  the  shores  are  such  that  it  is  quite 
impossible  to  determine  by  theory  what  the  course  and  character  of 
the  tide  wave  must  be.  We  must  depend  upon  observation ;  and 
observations  are  inadequate,  because,  with  the  exception  of  a  few 
islands,  our  only  possible  tide  stations  are  on  the  shores  of  continents 
where  local  circumstances  largely  control  the  phenomena. 

473.  Free  and  Forced  Oscillations.  —  If  the  water  in  the  ocean  is 
suddenly  disturbed  (as  for  instance,  by  an  earthquake),  and   then 
left  to  itself,  a  "  free"  wave  will  be  formed,  which,  if  the  horizontal 
dimensions  of  the  wave  are  large  as  compared  with  the  depth  of  the 
water,  will  travel  at  a  rate  depending  solely  on  the  depth.    The  veloc- 
ity of  such  a  free  wave  is  given  by  the  formula  v  =  ~\/gh ;  that  is,  it 
is  equal  to  the  velocity  acquired  by  a  body  in  falling  through  half  the 
depth  of  the  ocean. 

v 
Thus  a  depth  of    25  feet  gives  a  velocity  of     19  +  miles  per  hour. 

lOO      «  ..        u  u  u         ;j()  «  «          « 

10,000    "        "     "  "  "    388  "  "  " 

40,000    "        "     "  "  "    775  "  "  " 

67,200  (12|  miles)  "  "1000  «  «  « 

90,000    "        "     "  "  "  1165  "  "  " 

Observations  upon  the  waves  caused  by  certain  earthquakes  in 
South  America  and  Japan  have  thus  informed  us  that  between  the 
coasts  of  those  countries  the  Pacific  averages  between  two  and  one- 
half  and  three  miles  in  depth. 

474.  Now,  as  the  moon  in  its  diurnal  motion  passes  across  the 
American  continent  each  day,  and  comes  over  the  Pacific  Ocean,  it 
starts   such  a  "parent"  wave  in  the  Pacific,   and  the  wave  once 
started  moves  on  nearly  (but  not  exactly)  like  an  earthquake  wave. 
Not  exactly,  because  the  velocity  of  the  earth's  rotation  being  about 


310  THE  TIDES. 

1050  miles  an  hour  at  the  equator,  the  moon  runs  relatively  westward 
faster  than  the  wave  can  naturally  follow,  and  so  for  a  while  slightly 
accelerates  it.  A  second  tidal  wave  is  produced  daily  twelve  hours 
later  when  the  moon  passes  underneath.  The  tidal  wave  is  thus,  in 
its  origin ,  a  forced  oscillation ,  while  in  its  subsequent  travel  it  is  pretty 
nearly  a  free  one. 

475.  Co-Tidal  Lines. — These  are  lines   drawn  upon  the  surface 
of  the  ocean  connecting  those  places  which  have  their  high  water  at 
the  same  moment  of  Greenwich  time.     They  mark  the  crest  of  the 
tide  wave  for  each  hour  of  Greenwich  time ;  and  if  we  could  draw 
them  with   certainty  upon  the  globe,  we  should  have  all  necessary 
information  as  to  the  motion  of   the  wave.     Unfortunately  we  can 
obtain  no  direct  knowledge  as  to  the  position  of  these  lines  in  mid- 
ocean  ;  we  only  get  a  few  points  here  and  there  on  the  coasts  and 
on  islands,  so  that  a  great  deal  necessarily  remains  conjectural.    Fig. 
156  is  a  reduced  copy  of  such  a  map,  borrowed  with  some  modifica- 
tions from  that  given  in  Guyot's  u  Physical  Geography." 

476.  Course  of  Travel  of  the  Tidal  Wave.  —  On  studying  the  map 
we  find  that  the  main  or  "  parent "  wave  starts  twice  a  day  in  the  Pacific, 
off  Callao,  on  the  coast  of  South  America.     This  is  shown   on  the  chart 
by  a  sort  of  oval  "  eye  "  in  the  co-tidal  lines,  just  as  a  mountain  summit  is 
shown  on  a  topographical  chart  by  an  "  eye  "  in  the  contour  lines.    From 
this  point  the  wave  travels  northwest  through  the  deepest  water  of  the 
Pacific  at  the  rate  of  about  850  miles  per  hour,  reaching  Kamtchatka  in 
about  ten  hours.     To  the  west  and  southwest  the  water  is  shallower  and 
the  travel  slower,  —  only  400  to  600  miles  per  hour,  —  so  that  the  wave  arrives 
at  New  Zealand  in  about  twelve  hours.     Passing  on  by  Australia,  and  com- 
bining with  the  small  wave  which  the  moon  raises  directly  in  the  Indian 
Ocean,  the  resultant  tide  crest  reaches  the  Cape  of  Good  Hope  in  about 
twenty-nine  hours,  and  enters  the  Atlantic.     Here  it  combines  with  the  tide 
wave,  twenty-four  hours  younger,  which  has  "backed"  into  the  Atlantic 
around  Cape  Horn,  and  it  is  modified  also  by  the  direct  tide  produced  by  the 
moon's  action  upon  the  waters  of  the  Atlantic.     The  resultant  tide  crest 
then  travels  northward  through  the  Atlantic  at  the  rate  of  nearly  700  miles 
per  hour.     It  is  about  forty  hours  old  when  it  first  reaches  the  coast  of  the 
United  States  in  Florida,  and  our  coast  is  so  situated  that  it  arrives  at  all 
the  principal'  ports  within  two  or  three  hours  of  that  time.     It  is  forty-one 
or  forty-two  hours  old  when  it  arrives  at  New  York  and  Boston.     To  reach 
London  it  has  to  travel  around  the  northern  end  of  Scotland  and  through 
the  North  Sea,  and  is  nearly  sixty  hours  old  when  it  arrives  at  that  port  and 
the  ports  of  the  German  Ocean,  —  Hamburg,  etc. 

In  the  great  oceans  there  are  thus  three  or  four  tide  crests  travelling 


CO-TIDAL  LINES. 


311 


i\\)  3  W  lr^\<'^-^f  • 

!*Xr  m^^^ ^^L* i 
^  "Jt^i'-'i'//*-" — v..  .%-A  -    "~^  s 


s       s 


312  THE   TIDES. 

simultaneously,  following  each  other  nearly  in  the  same  track,  but  with 
continual  minor  changes,  owing  to  the  variations  in  the  relative  positions  of 
the  sun  and  moon  and  their  changing  distances  and  declinations.  If  we  take 
into  account  the  tides  in  rivers  and  sounds,  the  number  of  simultaneous 
tide  crests  must  be  at  least  six  or  seven  ;  that  is,  the  high  water  at  the 
extremity  of  its  travel,  up  the  Amazon  River,  for  instance,  must  be  at  least 
three  or  four  days  old,  reckoned  from  its  birth  in  the  Pacific.1 

477.  Tides  in  Rivers.  —  The  tide  wave  ascends  a  river  at  a  rate 
which  depends  upon  the  depth  of  the  water,  the  amount  of  friction, 
and  the  swiftness  of  the  stream.     It  may,  and  generally  does,  ascend 
until  it  comes  to  a  rapid,  where  the  velocity  of  the  water  is  greater  than 
that  of  the  wave.     In  shallow  streams,  however,  it  dies  out  earlier. 

Contrary  to  what  is  usually  supposed,  it  often  ascends  to  an  elevation  far 
above  that  of  the  highest  crest  of  the  tide  wave  at  the  river's  mouth.  In  the 
La  Plata  and  Amazon  it  goes  up  to  an  elevation  at  least  one  hundred  feet 
above  the  sea-level.  The  velocity  of  the  tide  wave  in  a  river  seldom  exceeds 
ten  or  twenty  miles  an  hour,  and  is  usually  less. 

478.  Height  of  Tides.  —  In  mid-ocean  the  difference  between  high 
and  low  water  is  usually  between  two  and  three  feet,  as  observed 
on  isolated  deep-water  islands  in  the  Pacific  ;  but  on  the  continental 
shores  the  height  is  usually  much  greater.     As  soon  as  the  tide  wave 


_ 

^=        ^    V  v  v 

PIG.  157.  —  Increase  in  Height  of  Tide  on  approaching  the  Shore. 

touches  bottom,  so  to  speak,  the  velocity  is  diminished  and  the  height 
of  the  wave  is  increased,  something  as  in  the  annexed  figure  (Fig. 
157).  Theoretically  the  height  varies  inversely  as  the  fourth  root  of 
the  depth.  Thus,  where  the  water  is  100  feet  deep,  the  tide  wave 
should  be  twice  as  high  as  at  the  depth  of  1600  feet. 

Where  the  configuration  of  the  shore  forces  the  wave  into  a  corner, 
it  sometimes  becomes  very  high.  At  the  head  of  the  Bay  of 
Fundy,  tides  of  seventy  feet  are  not  very  uncommon,  and  an  altitude 
of  nearly  a  hundred  feet  is  said  to  be  occasionally  attained. 

At  Bristol,  England,  in  the  mouth  of  the  Severn  the  tide  rises  fifty  feet, 
and  sometimes  ascends  the  river  (as  it  also  does  the  Seine,  in  France,  and 

1  We  are  greatly  indebted  to  Loomis's  discussion  of  the  subject  in  his  "Ele- 
ments of  Astronomy." 


REFLECTION    AND    INTERFERENCE.  313 

the  Amazon)  as  a  breaking  wave,  called  the  "bore"  or  "eiger"  (French. 
mascaret),  with  a  nearly  vertical  front  five  or  six  feet  in  height,  crested  with 
foam,  and  very  dangerous  to  small  vessels.  On  the  east  coast  of  Ireland, 
opposite  to  Bristol,  the  tide  ranges  only  about  two  feet. 

In  mid-ocean  the  water  has  no  progressive  motion,  but  near  the  land 
it  has,  running  in  at  the  flood  to  fill  up  the  bays  and  cover  the  flats,  and 
then  running  out  again  at  the  ebb.  The  velocity  of  these  tidal  currents  must 
not  be  confounded  with  that  of  the  tide  wave  itself. 

479.  Reflection  and  Interference.  —  The  tide  wave  when  it  reaches 
the  shore  is  not  entirely  destroyed,  especially  if  the  coast  is  bold  and 
the  water  deep  ;  but  is  partly  reflected,  and  the  reflected  wave  goes 
back  into  the  ocean  to  meet  and  modify  the  new  tide  wave  which  is 
coming  in.     Of  course,  in  such  a  case  we  get  "  interferences,"  so 
that  on  islands  in  the  Pacific  only  a  few  hundred  miles  apart  we  find 
great  differences  in  the  heights  of  the  tides.     At  one  place  the  direct 
waves  and  the  waves   reflected  from  the  shores  of  Asia  and  South 
America  may  conspire  to  give  a  tide  of  three  or  four  feet,  or  nearly 
double  its  normal  value,  while  at  another  they  nearly  destroy  each  other. 

There  are  places,  also,  which  are  reached  by  tides  coming  by  two  different 
routes.  Thus  on  the  east  coast  of  England  and  Scotland  the  tide  waves 
come  both  around  the  northern  end  of  Scotland  and  through  the  Straits  of 
Dover.  In  some  places  on  this  coast  we  have,  therefore,  a  tide  of  nearly 
double  height,  while  at  others  not  very  far  away  there  will  be  hardly  any 
tide  at  all;  and  at  intermediate  points  there  are  sometimes  four  distinct 
high  waters  in  twenty-four  hours.  As  a  consequence  of  this  reflection  and 
interference  of  the  tide  waves  it  follows  that  if  the  tide-raising  power  were 
suddenly  abolished,  the  tides  would  not  immediately  cease,  but  would  con- 
tinue to  run  for  several  days,  and  perhaps  weeks,  before  they  gradually  died 
out. 

480.  Effect  of  the  Varying  Pressure  of  the  Barometer,  and  of 
the  Wind.  —  When  the   barometer  at  a  given   port   is   lower   than 
usual,  the  level  of   the  water  is  generally  higher  than  the  average, 
at  the  rate  of  about  one  foot  for  every  inch  of  the  mercury  in  the 
barometer  ;  and  vice  versa  when  it  is  higher  than  usual. 

When  the  wind  blows  into  the  mouth  of  a  harbor,  it  drives  in 
the  water  of  the  ocean  by  its  surface  friction,  and  may  raise  the 
water  several  feet.  In  such  cases  the  time  of  high  water,  contrary 
to  what  might  at  first  be  supposed,  is  delayed,  sometimes  as  much  as 
fifteen  or  twenty  minutes. 

This  result  depends  upon  the  fact  that  the  water  runs  into  the 
harbor  for  a  longer  time  than  it  would  do  if  the  wind  were  not  blow- 


314  THE  TIDES. 

ing.  The  normal  depth  of  the  water  on  the  bar  is  reached  before 
the  predicted  time,  so  that  at  the  predicted  time  the  water  is  deeper 
than  it  would  be  if  there  were  no  wind,  but  the  maximum  depth  is 
not  attained  until  some  time  later.  Of  course,  the  results  are  the 
opposite  when  the  wind  blows  out  of  the  harbor :  the  time  of  high 
water  comes  earlier,  and  the  depth  of  water  on  the  bar  at  the  pre- 
dicted time  of  high  water  is  less  than  it  otherwise  would  be. 

481.  Tides  in  Lakes  and  Inland  Seas.  —  These  are  small  and  diffi- 
cult to  detect.     Theoretically,  the  range  between  high  and  low  water  in  a 
land-locked  sea  should  bear  about  the  same  ratio  to  the  rise  and  fall  of  the 
tide  in  mid-ocean  that  the  length  of  the  sea  does  to  the  diameter  of  the  earth. 
Variations  in  the  direction  of  the  wind  and  the  barometric  pressure  cause 
continual  oscillations 'in  the  water-level  which,  even  in  a  quiet  lake,  are  much 
larger  than  the  true  tides;  so  that  it  is  only  by  taking  a  long  series  of  obser- 
vations, and  discussing  them  with  reference  to  the  moon's  position  in  the  sky, 
that  it  is  possible  to  separate  the  real  tide  from  the  effects  of  other  causes. 
In  Lake  Michigan,  at  Chicago,  a  tide  of  about  one  and  three-quarters  inches 
has  thus  been  detected,  the  "  establishment "  of  the  port  being  about  thirty 
minutes.  In  Lake  Erie,  at  Buffalo  and  Toledo,  the  tide  is  about  three-quarters 
of  an  inch.     On  the  coasts  of  the  Mediterranean  the  tide  averages  about 
eighteen  inches,  attaining  a  height  of  three  or  four  feet  at  the  head  of  some 
of  the  bays. 

482.  The  Rigidity  of  the  Earth.  —  Lord  Kelvin  has  endeavored 
to  make  the  tides  the  criterion  of  the  rigidity  of  the  earth's  core. 
Evidently  if  the  solid  parts  of  the  earth  were  fluid,  there  would  be  no 
observable  tide  anywhere,  since  the  whole  surface  would  rise  and  fall 
together.     If  the  earth  were  semi-solid,  so  to  speak  (that  is,  viscous, 
and  capable  of  yielding  more  or  less  to  the  forces  tending  to  change 
its  form),  the  tides  would  be  observable,  but  to  a  less  degree  than  if 
the  earth's  core  were  rigid.    And  with  this  further  peculiarity  —  since 
a  viscous  body  requires  time  to  change  its  form,  waves  of  short  period 
would  be  observable  upon  the  semi-solid  earth  nearly  to  their  full  ex- 
tent,  while   those   of  long  period  would    almost   entirely  disappear, 
owing  to  the  slow  yielding  of  the  earth's  crust^  Now  the  actual  tide 
wave,  as  observed,  is  really  made  up  of  a  multitude  of  component 
tide  waves  of  different  periods,  ranging  from  half  a  day  upwards. 
According  to  the  "principle  of  forced  vibrations"  every  regularly 
recurring  periodic  change  in  the  forces  which  act  on  the  surface  of 
the  ocean  must  produce  a  tide  of  greater  or  less  magnitude,  and  of 
exactly  corresponding  period. 


EFFECT   OF  TIDES   ON  THE   EARTH'S   ROTATION.  315 

We  have,  for  instance,  the  semi-diurnal,  solar,  and  lunar  tides ;  then  the 
two  monthly  tides  due  to  the  change  in  the  moon's  distance  and  declination, 
and  the  two  annual  tides  due  to  the  changes  of  the  sun's  distance  and 
declination,  not  to  speak  of  the  nineteen-year  tide  due  to  the  revolution  of 
the  moon's  nodes. 

A  thorough  analytical  discussion  of  thirty-three  years'  tidal  ob- 
servations at  different  parts  of  the  world  has  been  made  under  the 
direction  of  Lord  Kelvin  by  Mr.  George  Darwin,  with  the  result 
that  not  only  do  the  short  waves  show  themselves,  but  the  waves  of 
long  period  are  found  to  manifest  themselves  with  almost  their  full 
theoretical  value.  Lord  Kelvin  concludes  that  the  earth  as  a  whole 
umust  be  more  rigid  than  steel,  but  perhaps  not  quite  so  rigid  as 
glass." *  This  result  is  at  variance  with  the  prevalent  belief  of  geolo- 
gists that  the  core  of  the  earth  is  a  molten  mass,  and  has  led  to  much 
discussion  which  we  cannot  deal  with  here. 

483.     Effect  of  the  Tides  on  the  Earth's  Rotation.— If  the  tidal 

motion  consisted  merely  in  the  upward  and  downward  motion  of  the 
particles  of  the  ocean  to  the  extent  of  two  feet  or  so  twice  a  day,  it 
would  involve  a  very  trifling  expenditure  of  energy ;  and  this  is  the 
case  with  the  mid-ocean  tide.  But  near  the  land  this  almost  insensi- 
ble mere  oscillatory  motion  is  transformed  into  the  bodily  travelling 
of  immense  masses  of  water,  which  flow  in  upon  the  shallows  and 
then  out  again  to  sea  with  a  great  amount  of  fluid  friction ;  and  this 
involves  the  expenditure  of  a  very  considerable  amount  of  energy 
which  is  dissipated  as  heat.  From  what  sources  does  this  energy 
come?  The  answer  is  that  it  must  be  derived  mainly  from  the 
earth's  energy  of  rotation,  and  the  necessary  effect  is  to  diminish  that 
energy  by  lessening  the  speed  of  the  rotation.  Compared  with  the 
earth's  whole  stock  of  rotational  energy,  however,  the  loss  of  it  by 
tidal  friction,  even  in  a  century,  is  very  small,  and  the  effect  on  the 
length  of  the  day  is  extremely  slight. 

The  reader  will  recall  the  remarks  upon  the  subject  of  the  secular  accel- 
eration of  the  moon's  mean  motion  a  few  pages  back  (Art.  461). 

While  it  is  certain  that  the  tidal  friction  tends  to  lengthen  the  day, 
it  does  not  follow  that  the  day  really  grows  longer.  There  are 
counteracting  causes :  —  for  example,  the  earth's  radiation  of  heat 
into  space,  and  the  consequent  shrinkage  of  her  volume. 

1  This  is  substantially  confirmed  by  recent  researches  of  S.  S.  Hough  upon 
the  variation  of  latitude. 


316 


THE   TIDES. 


As  matters  stand  we  do  not  know  whether,  as  a  fact,  the  day  is 
really  longer  or  shorter  than  it  was  a  thousand  years  ^rp.  The 
change,  if  any  has  really  occurred,  can  hardly  be  as  great  as  yoVir  °^ 
a  second. 

484.  Effect  of  the  Tide  on  the  Moon's  Motion.  —  Not  only  does 
the  tide  diminish  the  earth's  energy  of  rotation  directly  by  the  tidal 
friction,  but,  theoretically,  it  also  communicates 
a  minute  portion  of  that  energy  to  the  moon.  It 
will  be  seen  that  a  tidal  wave,  situated  as  in 
Fig.  158,  would  slightly  accelerate  the  moon's 
motion,  the  attraction  of  the  moon  by  the  tidal 
protuberance  F  being  slightly  greater  than  that 
of  the  tide  wave  at  I"  —  a  difference  tending  to 
increase  the  moon's  velocity  and  so  to  increase 
the  major  axis  of  its  orbit.  The  effect  is  there- 
fore to  make  the  moon  recede  from  the  earth,  and 
to  lengthen  the  month.  » 

Upon  this  interaction  between  the  tides  and 
the  motions  of  the  earth  and  moon  Professor 
George  Darwin  has  founded  his  theory  of  "tidal 
evolution"',  namely,  that  the  satellites  of  a  planet, 
Eff ect  of  the  Tide  on  the  having  separated  from  it  millions  of  years  ago, 

Moon's  Motion.  .         J 

have  been  made  to  recede  to  their  present  dis- 
tances by  just  such  an  action.  A  similar  action  is  invoked  by  Dr. 
See  to  explain  the  elongated  orbits  of  double  stars.  (Art.  877.) 

An  excellent  popular  statement  of  the  theory  will  be  found  in  the  closing 
chapter  of  Ball's  "  Story  of  the  Heavens,"  and  in  his  little  book  entitled 
"  Time  and  Tide."  The  original  papers  of  Mr.  Darwin  in  the  "  Philosophical 
Transactions"  are  of  course  severely  mathematical.^ 


FIG. 158. 


THE  PLANETS:    THEIR  MOTIONS.  317 


CHAPTER  XIV. 

THE  PLANETS :  THEIR  MOTIONS,  APPARENT  AND  REAL.  — 
THE  PTOLEMAIC,  TYCHONIC,  AND  COPERNICAN  SYSTEMS, 
—  THE  ORBITS  AND  THEIR  ELEMENTS.  —  PLANETARY 
PERTURBATIONS. 

485,  For  the  most  part,  the  stars  keep  their  relative  configurations 
unchanged,  however  much  they  alter  their  positions  in  the  sky  from 
hour  to  hour.     The  "  dipper"  remains  always  a  "  dipper"  in  what- 
ever part  of  the  heavens  it  may  be.     But  while  this  is  true  of  the 
stars  in   general,  certain  of  the  heavenly  bodies,   and  among  them 
those  that  are  the  most  conspicuous  of  all,  form  an  exception.     The 
sun   and    moon   continually    change   their  places,  moving   eastward 
among  the  stars ;  and  certain  others,  which  to  the  eye  appear  as  very 
brilliant  stars,  also  move,1  though  not  in  quite  so  simple  a  way. 

486.  These  bodies  were  named  by  the  Greeks  the  "planets" ;  that 
is,  "  wanderers."     The  ancient  astronomers  counted  seven  of  them. 
They  reckoned  the  sun  and  moon,  and  in  addition  Mercury,  Venus, 
Mars,  Jupiter,  and  Saturn. 

Venus  and  Jupiter  are  at  all  times  more  brilliant  than  any  of  the 
fixed  stars.  Mars  at  times,  but  not  usually,  is  nearly  as  bright  as 
Jupiter  ;  and  Saturn  is  brighter  than  all  but  a  very  few  of  the  stars. 
Mercury  is  also  bright,  but  seldom  seen,  because  always  near  the  sun. 

At  present  the  sun  and  moon  are. not  reckoned  as  planets;  but 
the  roll  includes,  in  addition  to  the  five  other  bodies  known  by  the 
ancients,  the  earth  itself,  which  Copernicus  showed  should  be  counted 
among  them,  and  also  two  new  bodies  of  great  magnitude  (though 
inconspicuous  because  of  their  distance)  which  have  been  discovered 
in  modern  times  ;  then  there  is  in  addition  a  host  of  so-called  "  aste- 
roids" which  circulate  in  the  otherwise  vacant  space  between  the 
planets  Mars  and  Jupiter. 

1  "When  we  speak  of  the  motion  of  the  planets,  the  reader  will  understand  that 
the  diurnal  motion  is  not  taken  into  account.  We  speak  of  their  motions  among 
the  stars. 


318  THE   PLANETS. 

487.  The  list  of  the  planets  in  the  order  of  distance  from  the  sun 
stands  thus  at  present :    Mercury,  Venus,  the  Earth,  Mars,  Jupiter, 
Saturn,  Uranus,  and  Neptune ;  and  between  Mars  and  Jupiter,  in  the 
place  where  a  planet  would  naturally  be  expected  to  revolve,  there 
are  at  present  known  about  700  little  planets,  which  probably  repre- 
sent a  single  one,  somehow  **  spoiled  in  the  making,"  so  to  speak,  or 
burst  into  fragments. 

The  planets  are  all  dark  bodies,  shining  only  by  reflected  sun- 
light, —  globes  which,  like  the  earth,  revolve  around  the  sun  in 
orbits  nearly  circular,  moving  all  in  the  same  direction,  and  (with 
some  exceptions  among  the  asteroids)  nearly  in  the  common  plane  of 
the  ecliptic  and  sun's  equator.  All  of  them  but  the  inner  two  and  the 
asteroids  are  also  attended  by  "  satellites."  Of  these  the  earth  has  one 
(the  moon),  Mars  two,  Jupiter  seven,  Saturn  ten,1  Uranus  four,  and 
Neptune  one  ;  i.e.,  so  far  as  at  present  known  ;  for  although  it  is  hardly 
probable,  it  is  not  at  all  impossible  that  others  may  yet  be  found. 

488.  Relative  Distances  of  Planets  from  the  Sun:  Bode's  Law. 
—  There  is  a  curious  approximate   relation   between  the  distances 
of  the  planets  from  the  sun,  which  makes  it  easy  to  remember  them. 
It  is  usually  known  as  Bode's  Law,  because  Bode  first  brought  it 
prominently  into  notice  in  1772,  though  Titius  of  Wittenberg  seems 
to  have  discovered  and  enunciated  it  some  years  earlier.     The  law  is 
this  :  Write  a  series  of  4's.    To  the  second  4  add  3  ;  to  the  third  add 
3  X  2,  or  6  ;  to  the  fourth,  3x4,  or  12  ;  and  so  on,  doubling  the 
added  number  each  time,  as  in  the  accompanying  scheme. 


4 
4 

4 
3 

7 
9 

4 
6 
10 

e 

4 
12 
16 

4 

24 

4 

48 

4 

96 

4 
192 
196 
¥ 

4 

384 
388 

[28] 

52 

100 

h 

'  The  resulting  numbers,  divided  by  10,  are  pretty  nearly  the  true 
mean  distances  of  the  planets  from  the  sun,  in  terms  of  the  radius 
of  the  earth's  orbit.  In  the  case  of  Neptune,  however,  the  law 
breaks  down  utterly,  and  is  not  even  approximately  correct. 

For  the  present,  at  least,  the  law  is  to  be  regarded  as  a  mere 
coincidence.  Its  explanation  may  perhaps  ultimately  be  found  in 
the  process  by  which  the  system  was  developed. 

The  general  expression  for  the  nth  term  of  the  series  is  4  +  3  X  2<w~2>; 
but  it  does  not  hold  good  of  the  first  term,  which  is  simply  4,  instead  of 
being  5£,  i.e.,  (4  +  3  x  2-1),  as  it  should  be. 

1  See  note  on  page  406. 


TABLE  OF  NAMES,  DISTANCES,   AND   PERIODS. 


319 


489.     Table  of  Names,  Distances,  and  Periods. 


NAME. 

SYMBOL. 

DISTANCE. 

BODE. 

DlFF. 

SID.  PERIOD. 

SYN. 
PERIOD. 

Mercury  .... 
Venus      .      .  . 

0 

0.387 
0.723 

0.4 
0.7 

-0.013 
+  0.023 

88d     or        3m 
224.7d  or      7^m 

116d 

584d 

Earth    ..... 

1.000 

1.0 

0.000 

365^d  or  ly 

$ 

1.523 

1.6 

—  0.077 

687d     or  ly  10  lm 

780d 

Mean  Asteroid 

2.650 

2.8 

-0.150 

3y.l  to  8y.O 

various 

Jupiter    .... 
Saturn 

y 

5.202 
9.539 

5.2 
100 

+  0.002 
—  0461 

lly.9 

399d 
378d 

Uranus    .... 
Neptune  .... 

$&¥ 
V 

19.183 
30.054 

19.6 

38.8 

-0.417 
-8.746! 

84y.O 
164y.8 

370d 

The  column  headed  "  Bode  "  gives  the  distance  according  to  Bode's  law ;  the 
column  headed  "  Din0.,"  the  difference  between  the  true  distance  and  that  given  by 
Bode's  law. 


FlO.  159.  —  Plan  of  the  Orbits  of  the  Planets  inside  of  Saturn. 


320  THE    PLANETS. 

490.  Fig.  159  shows  the  smaller  orbits  of  the  system  (including  the  orbit 
of  Jupiter)  drawn  to  scale,  the  radius  of  the  earth's  orbit  being  taken  as  one 
centimetre.     On  this  scale  the  diameter  of  Saturn's  orbit  would  be  19cm.08, 
that  of  Uranus  would  be  38cm.36,  and  that  of  Neptune,  60cm.ll.      The 
nearest  fixed  star  on  the  same  scale  would  be  about  a  mile  and  a  quarter 
away.     The  dotted  half  of  each  orbit  is  that  which  lies  below,  i.e.,  south  of, 
the  plane  of  the  ecliptic.     The  place  of  perihelion  of  each  planet's  orbit  is 
marked  with  a  P.     The  orbits  of  five  of  the  asteroids  are  also  given. 

491.  Periods.  —  The  sidereal  period  of  a  planet  is  the  time  of  its 
revolution  around  the  sun  from  a  star  to  the  same  star  again,  as  seen 
from  the  sun.     The  synodic  period  is  the  time  between  two  successive 
conjunctions  of  the  planet  with  the  sun,  as  seen  from  the  earth.    The 
sidereal  and  synodic  periods  are  connected  by  the  same  relation  as 
the  sidereal  and  synodic  months  (Art.  232)  ;  namely,  — 


in  which  E,  P,  and  S  are  respectively  the  periods  of  the  earth  and  of 
the  planet,  and  the  planet's  synodic  period,  and  the  numerical  differ- 

ence between  —  and  —  is  to  be  taken  without  regard  to  sign. 
_t  xL 

—  is  the  planet's  mean  daily  motion  expressed  as  a  fraction  of  the  whole 
circumference,  while  —  is  the  earth's  motion;    and  the  equation  simply 


states  that  the  daily  synodic  motion  (^J>  is  the  difference  of  the  other  two 

motions.     The  two  last  columns  of  the  table  in  Article  489  give  the  approxi- 
mate periods,  both  sidereal  and  synodic,  for  the  different  planets. 

492.  Apparent  Motions.  —  As  viewed  from  a  distant  point  on  the 
line  drawn  through  the  sun,  perpendicular  to  the  plane  of  the  eclip- 
tic, the  planets  would  be  seen  to  travel  in  their  nearly  circular  orbits 
with  a  regular  motion.    As  seen  from  the  earth  the  apparent  motion 
is  much  more  complicated,  being  made  up  of  their  real  motion  around 
the  sun  combined  with  an  apparent  motion  due  to  the  earth's  own 
movement. 

493.  Law  of  Relative  Motion.  —  The  motion  of  a  body  relative 
to  the  earth  can  be  very  simply  stated.     It  is  always  the  sam,e  as  if 
the  body  had,  combined  with  its  own  motion,  another  motion,  identical 
with  that  of  the  earth,  but  reversed. 


COMBINATION   OF   EARTH'S    AND   PLANET'S   MOTIONS.     321 


The  proof  of  this  is  simple.  Let  E,  Fig.  160,  be  the  earth,  and  P  the 
planet,  its  direction  and  distance  being  given  by  the  line  EP.  Let  E  have 
a  motion  which  will  take  it  to  E'  in  a  unit  of  time,  and  P  a  motion  which 
will  take  it  to  P'  in  the  same  time.  Then  at  the  end  of  a  unit  of  time  the 
distance  and  direction  of  P  from  E  will  be  given  by  the  line  E'P'.  But  if 
we  suppose  E  to  remain  at  rest,  and  give  to  P  a  motion  Pe  equal  to  EE'  but 

E' 


FIG.  160.  —  The  Relative  Motions  of  Two  Bodies. 

opposite  in  direction,  and  combine  this  motion  with  PP'  by  drawing  the 
parallelogram  of  motions,  we  shall  get  P"  for  the  resulting  place  of  P  at 
the  end  of  the  unit  of  time ;  and  because  the  line  EP"  is  parallel  and  equal 
to  E'P'  (as  follows  from  the  construction),  the  point  P",  as  seen  from  Ey 
would  occupy,  in  the  celestial  sphere,  precisely  the  same  position  as  P'  seen 
from  E' ;  since  all  parallel  lines  pierce  the  sphere  at  one  and  the  same  opti- 
cal point  (Art.  7). 

If,  therefore,  the  earth  moves  in 
a  circle,  every  body  really  at  rest 
will  appear  to  move  in  a  circle  of 
the  same  size  as  the  earth's  orbit, 
but  keeping  in  such  a  part  of  its 
circle  as  always  to  have  its  motion 
precisely  opposite  to  the  earth's 
own  real  motion  at  the  moment. 
We  shall  have  occasion  to  use  this 
principle  very  frequently. 

494.  Geocentric  Motion  of  a 
Planet  in  Space. — The" geocentric" 

motion  of  a  planet  is  its  motion 

relative  to  the  earth  regarded  as  a 

fixed  centre,  and  is  therefore  made  up  of  two  motions,  —  that  of  a 

body  moving  once  a  year  around  the  circumference  of  a  circle  equal 


FIG.  161. 

Geocentric  Motion  of  Jupiter  from  1708  to 
1720.    (Cassini.) 


322 


THE   PLANETS. 


to  the  earth's  orbit,  while  the  centre  of  this  circle  itself  goes  around 
the  sun  upon  the  real  orbit  of  the  planet,  and  with  a  periodic  time 
equal  to  that  of  the  planet.  In  other  words,  its  distance  and  direction 
from  the  earth  are  always  just  what  they  would  be  if  the  earth  were 
at  rest  while  the  planet  itself  moved  in  this  complicated  manner.  As 
the  result  of  this  combination  of  motions  the  relative,  or  "geocentric," 
orbit  of  a  planet  is  a  looped  curve,  which,  if  the  real  orbits  were  per- 
fectly circular,  would  be  exactly  an  epicycloid.  Since,  however,  the 
orbits  are  slightly  oval  the  loops  actually  vary  somewhat  in  size  and 
distance  from  each  other.  Jupiter,  for  instance,  appears  to  move  as 
in  Fig.  161,  making  eleven  loops  in  each  revolution,  the  smaller 
circle  having  a  diameter  of  about  one-fifth  that  of  the  larger  one, 
upon  which  its  centre  moves,  since  the  diameter  of  Jupiter's  orbit 
is  about  five  times  that  of  the  earth's.  (For  fuller  illustration  see 
Appendix,  Art.  1009.) 

495,    Apparent  Motions  of  the  Planets  upon  the  Celestial  Sphere. 
Direct  and  Retrograde  Motions  and  Stationary  Points.  --  As    a 

Superior  Conjunction 


Opposition 
FIG.  162.  —  Planetary  Configurations  and  Aspects. 

consequence  of  this  looped  motion  we  have  a  peculiar  back-and-forth 
movement  of  the  planets  among  the  stars.  Starting  from  the  time 
when  the  sun  is  between  us  and  the  planet,  —  the  time  of  superior 


MOTIONS    OF   THE   PLANETS.  323 

conjunction,1  as  it  is  called,  because  the  planet  is  then  above  the  sun, 
i.e.,  further  from  the  earth  (at  a  in  its  geocentric  orbit,  Fig.  161),— 
«  the  planet  moves  eastward  among  the  stars  for  a  certain  time,  con- 
tinually increasing  its  longitude  (and  also  its  right  ascension)  until 
at  last  its  apparent  motion  slackens  and  it  becomes  stationary  at  the 
points  marked  c,  c,  c,  in  the  geocentric  orbit,  Fig.  161.  ^XJbie^  elonga- 
tion of  this  stationary  point  from  the  sun  depends  upon  the  size  of 
the  planet's  orbit  compared  with  that  of  the  earth. 

Then  it  reverses  its  motion  and  moves  westward,  or  "retrogrades," 
for  a  while,  the  middle  of  the  arc  of  retrogression  being  passed  at  the 
time  when  the  earth  and  planet  are  in  line  with  the  sun,  and  on  the 
same  side  of  it  at  the  points  marked  b,  b,  in  the  geocentric  orbit,  — 
the  ends  of  the  "loops'7  where  the  distance  of  the  planet  from  the 
Earth  is  a  minimum.  If  the  planet  is  one  of  the  outer  ones,  it  will 
then  be  opposite  to  the  sun  in  the  sky  like  the  full  moon,  and  is  said 
to  be  "in  opposition."  If  it  is  one  of  the  inferior  planets  (Venus  or 
Mercury),  it  will  then  be  in  "inferior  conjunction"  as  it  is  called, 
between  the  earth  and  sun. 

After  the  planet  has  completed  its  arc  of  retrogression,  it  again 
becomes  stationary,  turns  upon  its  course,  and  once  more  advances 
eastward  among  the  stars,  until  the  synodic  period  is  completed  by 
its  re-arrival  at  superior  conjunction. 

Both  in  the  number  of  degrees  passed  over,  and  in  the  time  spent 

in  this  motion,  the  eastward  or  "direct"  motion  always  exceeds  the 

retrograde.     In  the  case  of  the  remoter  planets  the  excess  is  small 

-  from  3°  to  10° ;  in  the  case  of  the  nearest  ones,  Mars  and  Venus, 

it  is  many  times  greater. 

X  As  observed  with  a  sidereal  clock,  all  the  planets  come  later  to  the 
meridian  each  night  when  moving  direct,  since  their  right  ascension  is 
then  increasing  ;  but  vice  versa,  of  course,  when  they  are  retrograding. 

496.  Motions  in  Latitude.  —  If  their  orbits  lay  precisely  in  the 
same  plane  as  that  of  the  earth,  the  planets  would  always  keep 
exactly  upon  the  ecliptic.  In  fact,  however,  they  deviate  from  that 


1  We  give  Fig.  162  to  illustrate  the  meaning  of  the  different  terms,  Opposition, 
Quadrature,  Inferior  and  Superior  Conjunction,  and  Greatest  Elongation.  E  is 
the  position  of  the  earth,  the  inner  circle  being  the  orbit  of  an  inferior  planet, 
while  the  outer  circle  is  the  orbit  of  a  superior  planet.  In  general,  the  angle  PES 
(the  angle  at  the  earth  between  lines  drawn  from  the  earth  to  the  planet  and  to 
the  sun)  is  the  planet's  elongation  at  the  moment.  For  a  superior  planet  it  can 
have  any  value  from  zero  to  180°;  for  an  inferior  it  has  a  maximum  value  that 
the  planet  cannot  exceed,  depending  upon  the  diameter  of  its  orbit. 


324 


THE    PLANETS. 


circle  to  the  extent  of  4°  or  5°,  and  Mercury  sometimes  as  much  as 
8°;  so  that  their  paths  among  the  stars  form  loops  and  kinks.     Fig. 


Fig.  163.  —  Motion  of  Saturn  and  Uranus  in  1897. 

163  shows,  as  an  example,  the  apparent  motions  of  Saturn  and 
Uranus  for  1897.  In  the  case  of  Mars  the  loops  are  usually  much 
more  intricate. 

497.  Motion  of  the  Planets  with  Respect  to  the  Sun's  Place  in 
the  Sky.    Change  of  Elongation.  —  The  visibility  of  a  planet  depends 
mainly  upon  its  angular  distance,  or  "elongation"  from  the  sun, 
because  when  near  the  sun  the  planet  will  be  above  the  horizon  only 
by  day,  and  cannot  usually  be  seen.     As  regards  their  motions,  con- 
sidered from  this  point  of  view,  there  is  a  marked  difference  between 
the  inferior  planets  and  the  superior. 

498.  Behavior  of  a  Superior  Planet.  —  The  superior  planets  drop 
always  steadily  westward  with  respect  to  the  sun's  place  in  the  heavens, 
continually  increasing  their  western  elongation,  or  decreasing  their 
eastern :  they  therefore  invariably  come  earlier  to  the  meridian  every 
successive  night,  as  observed  by  a  time-piece  keeping  solar  time. 

Beginning  at  superior  conjunction,  the  planet  is  then  moving  eastward 
among  the  stars  with  its  greatest  speed  ;  but  even  then  its  eastward  motion 
is  not  so  great  as  the  sun's,  and  so  the  planet  relatively  falls  westward.  After 
a  while  it  will  have  fallen  behind  by  90°,  and  will  then  be  in  western  quad- 
rature, and  on  the  meridian  at  sunrise ;  at  the  end  of  half  its  synodic  period 
it  will  have  lost  180°,  and  will  be  just  opposite  the  sun  at  sunset,  being  then 
at  its  least  possible  distance  from  the  earth,  and  at  its  greatest  brilliance. 
At  this  time  the  difference  between  the  times  of  its  daily  culminations  is 


MOTIONS    OF    THE    PLANETS.  325 

also  the  greatest  possible,  and  may  be  as  much  (in  the  case  of  Mars)  as  six 
minutes,  by  which  amount  it  arrives  at  the  meridian  earlier  each  successive 
night.  After  opposition  the  planet  is  higher  in  the  sky  each  night  at  sunset 
until  it  reaches  eastern  quadrature,  when  it  is  90°  east  of  the  sun,  and  there- 
fore on  the  meridian  at  sunset.  Thence  it  drops  back,  falling  more  and 
more  slowly  westwards  towards  the  sun,  until  the  synodic  period  is  com- 
pleted by  a  new  conjunction.  $ 

499.  Motion  of  an  Inferior  Planet. — The  inferior  planets  appear, 
on  the  other  hand,  to  vibrate  across  the  sun,  moving  out  equal  dis- 
tances on  each  side  of  it,  but  making  the  westward  swing  much 
quicker  than  the  eastern. 

The  reason  of  this  difference  is  obvious  from  Fig.  162.  Matters 
take  place  with  respect  to  the  earth,  sun,  and  planet  as  if  the  earth 
were  at  rest  and  the  planet  revolving  around  the  sun  once  in  a  syn- 
odic (not  sidereal)  period.  Now,  since  the  distance  between  the 
points  of  greatest  elongation,  Fand  F',  is  less  through  inferior  con- 
junction 7,  than  from  V  around  to  V  through  (7,  the  time  ought  to 
be  correspondingly  shorter,  as  it  is. 

500.  The  Ptolemaic  System.  —  The  ancient  astronomers,  for  the 
most  part,  never  doubted  the  fixity  of  the  earth,  and  its  position  in 
the  centre  of  the  celestial  universe,  though  there  are  some  reasons  to 
think  that  Pythagoras  may  have  done  so.     Assuming  this  and  the 
actual  diurnal  revolution  of  the  heavens,  Ptolemy,  who  flourished  at 
Alexandria  about  140  A.D.,  worked  out  the  system  which  bears  his 
name.    His  MeyaA??  5iWa£i?  (or  Almagest  in  Arabic)  was  for  fourteen 
centuries  the  authoritative  "Scripture  of  Astronomy."    *He  showed 
that  all  the  apparent  motions  of  the  planets  could  be  accounted  for 
by  supposing  each  planet  to  move  around  the  circumference  of  a 
circle  called  the  "epicycle"  while  the  centre  of  this  circle,  sometimes 
called  the  "fictitious  planet"  itself  moved  on  the  circumference  of 
another  and  larger  circle  called  the  "deferent"     It  was  as  if  the 
real  planet  was  carried  on  the  end  ~bf  a  crank-arm  which  turned 
around  the  fictitious  planet  as  a  centre,  in  such  a  way  as  to  point 
towards  or  from  the  earth  at  times  when  the  planet  is  in  line  with 
the  sun. 

In  the  case  of  the  superior  planets  the  revolution  in  the  epicycle  was 
made  once  a  year,  so  that  the  "crank-arm"  was  always  parallel  to  the  line 
joining  earth  and  sun,  while  the  motion  around  the  deferent  occupied  what 
we  now  call  the  planet's  period.  Fig.  164  represents  the  Ptolemaic  System, 
except  that  no  attention  is  paid  to  dimensions,  the  "deferents"  being  spaced 


326 


THE  PLANETS. 


at  equal  distances.  It  will  be  noticed  that  the  epicycle-radii  which  carry  at 
their  extremities  the  planets  Mars,  Jupiter,  and  Saturn  are  all  always  parallel 
to  the  line  that  joins  the  earth  and  sun.  In  the  case  of  Venus  and  Mercury 
this  was  not  so.  Ptolemy  supposed  that  the  deferent  circles  for  these  planets 
lay  between  the  earth  and  the  sun,  and  that  the  "fictitious  planet"  in  both 


FIG.  164.  —  The  Ptolemaic  System. 

cases  revolved  in  the  deferent  once  a  year,  always  keeping  exactly  between 
the  earth  and  the  sun  :  the  motion  in  the  epicycle  in  this  case  was  completed 
in  the  time  of  the  planet's  period,  as  we  now  know  it.  He  ought  to  have 
seen  that,  for  these  two  planets,  the  deferent  was  really  the  orbit  of  the 
sun  itself,  as  the  ancient  Egyptians  are  said  to  have  understood. 

501.  To  account  for  some  of  the  irregularities  of  the  planets'  motions 
it  was  necessary  to  suppose  that  both  the  deferent  and  epicycle,  though 
circular,  are   eccentric,  the   earth   not  being  exactly  in  the  centre  of  the 
deferent,  nor  the  "  fictitious  planet "  in  the  exact  centre  of  the   epicycle. 
In  after  times,  when  the  knowledge  of  the  planetary  motions  had  become 
more  accurate,  the  Arabian  astronomers  added  epicycle  upon  epicycle  until 
the  system  became  very  complicated.     King  Alphonso  of  Spain  is  said  to 
have  remarked  to  the  astronomers  who  presented  to  him  the  Alphonsine 
tables  of  the  planetary  motions,  which  had  been  computed  under  his  orders, 
that  "  if  he  had  been  present  at  the  creation  he  would  have  given  some  good 
advice." 

502.  Some  of  the  ancient  astronomers  attempted  to  account  for  the  plan- 
etary and  stellar  motions  in  a  mechanical  way  by  means  of  what  were  called 
the  " crystalline  spheres"     The  planet  Jupiter,  for  instance,  was  supposed  to 


THE    COPERNICAN    SYSTEM.  327 

be  set  like  a  jewel  on  the  surface  of  a  small  globe  of  something  like  glass,  and 
this  itself  was  set  in  a  hollow  made  to  fit  it  in  the  thick  shell  of  a  still  larger 
sphere  which  surrounded  the  earth.  Thus  the  planets  were  supported  and 
carried  by  the  motions  of  these  invisible  crystalline  spheres ;  but  this  idea, 
though  prevalent,  was  by  no  means  universally  accepted. 

503.  Copernican  System, — Copernicus  (1473-1543)  asserted  the 
diurnal  rotation  of  the  earth  on  its  axis,  and  showed  that  it  would 
fully  account  for  the  apparent  diurnal  revolution  of  the  stars.     He 
also  showed  that  nearly  all  the  known  motions  of  the  planets  could 
be  accounted  for  by  supposing  them  to  revolve  around  the  sun,  with 
the  earth  as  one  of  them,  in  orbits  circular,  but  slightly  out  of  centre. 
His  system,  as  he  left  it,  was  nearly  that  which  is  accepted  to-day, 
and  Fig.  159  may  be  taken  as  representing  it.     He  was,  however, 
obliged  to  retain  a  few  small  epicycles  to  account  for  certain  of  the 
irregularities. 

So  far,  no  one  dared  to  doubt  the  exact  circularity  of  celestial 
orbits.  It  was  metaphysically  improper  that  heavenly  bodies  should 
move  in  any  but  perfect  curves,  and  the  circle  was  regarded  as  the 
only  perfect  one.  It  was  left  for  Kepler,  some  sixty-five  years  later 
than  Copernicus,  to  show  that  the  planetary  orbits  are  elliptical,  and  to 
bring  the  system  substantially  into  the  form  in  which  we  know  it  now. 

504.  Tychonic  System.  —  Tycho  Brahe,  who  came  between  Copernicus 
and  Kepler,  found  himself  unable  to  accept  the  Copernican  system  for  two 
reasons.     One  reason  was  that  it  was  unfavorably  regarded  by  the  clergy, 
and  he  was  a  good  churchman.     The  other  was  the  scientific  objection  that 
if  the  earth  moved  around  the  sun,  thejlxed  stars  all  ought  to  appear  to  move 
in  a  corresponding  manner  (Art.  492),  each  star  describing  annually  an  oval 
in  the  heavens  of  the  same  apparent  dimensions  as  the  earth's  orbit  itself, 
seen  from  the  star.     Technically  speaking,  they  ought  to  have  an  "annual 
parallax."     His  instruments  were  by  far  the  most  accurate  that  had  so  far 
been  made,  and  he  could  detect  no  such  parallax ;  hence  he  concluded,  not 
illogically,  but  incorrectly,  that  the  earth  must  be  at  rest.     He  rejected  the 
Copernican  system,  placed  the  earth  at  the  centre  of  the  universe,  according 
to  the  then  received  interpretation  of  Scripture,  made  the  sun  revolve  around 
the  earth  once  a  year,  and  then  (this  was  the  peculiarity  of  his  system)  made 
all  the  planets  except  the  earth  revolve  around  the  sun. 

This  theory  just  as  fully  accounts  for  all  the  motions  of  the  planets  as 
the  Copernican,  but  breaks  down  absolutely  when  it  encounters  the  aberra- 
tion of  light,  and  the  annual  parallax  of  the  stars,  which  we  can  now  detect 
with  our  modern  instruments,  although  Tycho  could  not  with  his.  The 
Tychonic  system  never  was  generally  accepted ;  the  Copernican  was  very 
soon  firmly  established  by  Kepler  and  Newton. 


328  THE   PLANETS. 

505,  Elements  of  a  Planet's  Orbit.  —  These  are  certain  numerical 
quantities  which    describe  the  orbit  with   precision,   and   furnish 
the  means  of  finding  the  planet's  place  in  the  orbit  at  any  given 
time,  whether  past  or  future,  so  far  as  that  place  depends  upon  the 
attraction  of  the  sun  alone.     Those  usually  employed  are  seven  in 
number,  as  follows  :  — 

1.  The  semi-major  axis,  a. 

2.  The  eccentricity,  e. 

3.  The  inclination  to  the  ecliptic,  i. 

4.  The  longitude  of  the  ascending  node,  ££. 

5.  The  longitude  of  perihelion,  IT. 

6.  The  epoch,  E. 

7.  The  period  P,  or  daily  motion  /A. 

506.  Of  these,  the  first  five  pertain  to  the  orbit  itself,  regarded  as 
an  ellipse  lying  in  space  with  one  focus  at  the  sun,  while  two  are 
necessary  to  determine  the  planet's  place  in  the  orbit. 

The  semi-major  axis,  a  (CA  in  Fig.  165),  defines  the  Size  of  the 

B 

M 


FIG.  165. —  The  Elements  of  a  Planet's  Orbit. 

orbit,  and  may  be  expressed  either  in  "astronomical  units"  (the 
earth's  mean  distance  from  the  sun  is  the  astronomical  unit)  or  in 
miles. 

The  Eccentricity  defines  the  orbit's  Form.     It  is  a  mere  numerical 

quantity,  being  the  fraction  -  (usually  expressed  decimally),  obtained 

by  dividing  the  distance  between  the  sun  and  the  centre  of  the  orbit 
by  the  semi-major  axis.  In  some  computations  it  is  convenient  to 
use,  instead  of  the  decimal  fraction  itself,  the  angle  </>  which  has  e  for 
its  sine  ;  i.e.,  <f>  =  sin~Je. 


ELEMENTS    OF    A    PLANET'S    ORBIT.  329 

The  third  element,  i,  is  the  Inclination  between  the  plane  of  the 
planet's  orbit  and  that  of  the  earth.  In  the  figure  it  is  the  angle 
KNO,  the  plane  of  the  ecliptic  being  lettered  EKLM. 

The  fourth  element,  ^  (the  Longitude  of  the  ascending  node), 
defines  what  has  been  called  the  "aspect"  of  the  orbit-plane  ;  i.e., 
the  direction  in  which  it  faces.  The  line  of  nodes  is  the  line  NN'  in 
the  figure,  the  intersection  of  the  two  planes  of  the  orbit  and  ecliptic ; 
and  the  angle  TSN  is  the  longitude  of  the  ascending  node,  the 
line  $T  being  the  line  drawn  from  the  sun  to  the  first  of  Aries.  The 
planet,  moving  around  its  orbit  in  the  plane  ORBT,  and  in  the  direc- 
tion of  the  arrow,  passes  from  the  lower  or  southern  side  of  the  plane 
of  the  ecliptic  to  the  northern  at  the  point  n,  which,  as  seen  from  S, 
is  in  the  same  direction  as  N. 

The  fifth,  and  last,  of  the  elements  which  belong  strictly  to  the 
orbit  itself  is  ?r,  the  so-called  Longitude  of  the  perihelion,  which  de- 
fines the  direction  in  which  the  major  axis  of  the  ellipse  (the  line  PA) 
lies  on  the  plane  ORBT.  It  is  not  strictly  a  longitude,  but  equals 
the  sum  of  the  two  angles  ££  anc^  w  »  **•£•»  ^SN  (in  the  plane  of  the 
ecliptic)  -j-  NSP  (in  the  plane  of  the  orbit  and  reckoned  in  the 
direction  of  the  planet's  motion).  NSP  exceeds  180°  in  the  figure. 
It  is  quite  sufficient  to  give  o>  alone,  and  in  the  case  of  cometary 
orbits  this  is  usually  done. 

507.  If  we   regard  the  orbit  as  an  oval  wire  hoop  suspended 
in  space,  these  five  elements  completely  define  its  position,  form, 
and  size.     The  plane  of  the  orbit  is  fixed  by  the  elements  numbered 
three  and  four,  the  position  of  the  orbit  in  this  plane  by  number 
five,  the  form  of  it  by  number  two,  and  finally  its  magnitude  by 
number  one. 

508.  To  determine  where  the  planet  will  be  at  any  subsequent 
date  we  need  two  more  elements. 

Sixth.  The  Periodic  Time,  —  we  must  have  the  sidereal  period,  P, 
or  else  the  mean  daily  motion,  p,  which  is  simply  360°  divided  by 
the  number  of  days  in  P. 

Seventh.  And  finally ;  we  must  have  a  starting-point,  the  "JSpoch" 
so-called ;  i.e.,  the  longitude  of  the  planet  as  seen  from  the  sun,  at 
some  given  date,  usually  Jan.  1st,  1850  or  1900,  or  else  some  precise 
date  at  which  the  planet  passed  the  perihelion  or  node. 

509.  If  it  were  not  for  perturbations  caused  by  the  mutual  interaction 
between  the  planets,  these  elements  would  never  change,  and  could  be  used 


330  THE   PLANETS. 

directly  for  computing  the  planet's  place  at  any  date  in  the  past  or  in  the 
future ;  but,  excepting  a  and  P,  they  do  change  on  account  of  such  interac- 
tion and  accordingly  it  is  usual  to  add  in  tables  of  the  planetary  elements, 
columns  headed  A&,  ATT,  A»,  and  Ae,  giving  the  amount  by  which  the 
quantities  &,  TT,  t,  and  e  respectively  change  in  a  century. 

510.  If  Kepler's  harmonic  law  were  strictly  true,  we  should  not  need 
both  a  and  P,  because  we  should  have 

(Earth's  Period)2 :  P2 : :  I3  :  a3,  or  P  =  at, 

P  being  expressed  in  years  and  a  in  astronomical  units.  But  since  the  exact 
form  of  the  equation  is 

P,2(l  +  mx)  :  P22(l  +  m2)  ::  a,s  :  a*  (Art.  417), 
it  is  necessary  to  regard  P  and  a  as  independent,  and  give  both  of  them. 

511.  Geocentric  Place.  —  Our  observations  of  a  planet's  place  are 
necessarily  "geocentric"  or  earth-centred ;  they  give  us,  when  prop- 
erly corrected  for  refraction  and  parallax,  the  planet's  right  ascension 
and  declination  as  seen  from  the  centre  of  the  earth,  and  from  them, 
if  desired,  the  corresponding  geocentric  longitude  and  latitude  are 
easily  obtained  by  the  method  explained  in  Article  180. 

511*.  Interpolation  of  Observations.  —  It  often  happens  that  we 
want  the  place  at  some  moment  of  time  when  the  planet  could  not 
be  directly  observed,  as,  for  instance,  in  the  day  time.  If  we  have 
a  series  of  observations  of  the  planet  made  about  that  time,  the  place 
for  the  exact  moment  is  readily  deduced  by  a  process  of  interpolation, 
and  with  an  accuracy  actually  exceeding  that  of  any  single  observa- 
tion of  the  series. 

Graphically  it  is  done  by  simply  plotting  the  observations  actually  made. 
Suppose,  for  instance,  we  want  the  right  ascension  of  Mars  for  8  A.M.  on 
June  3,  and  have  meridian-circle  observations  made  at  10  o'clock  P.M.  on 
June  1,  at  9h  55m  on  June  2,  at  9h  50m  on  June  3,  and  so  on.  We  first  lay 
off  the  times  of  observation  as  abscissas  along  a  horizontal  line  taken  as  the 
time-scale,  and  then  lay  off  the  observed  right  ascensions  as  ordinates  at 
points  corresponding  to  the  times.  Then  we  draw  a  smooth  curve  through 
the  points  so  determined,  and  from  this  curve  we  can  read  off  directly,  the 
right  ascension  corresponding  to  any  desired  moment.  The  declination  can 
be  treated  in  the  same  way.  Of  course,  what  can  be  done  graphically  can 
be  done  still  more  accurately  by  computation. 

512.  Heliocentric  Place. — The  heliocentric  place  of  a  planet  is  the 
place  as  seen  from  the  sun ;  and  when  we  know  the  longitude  of  the 
node  of  a  planet's  orbit  and  its  inclination,  as  well  as  the  planet's  dis- 


DETERMINATION   OF   A   PLANET'S   PERIOD.  331 

tance  from  the  sun,  this  heliocentric  place  can  at  once  be  deduced 
from  the  geocentric  by  a  trigonometrical  calculation.  The  process  is 
rather  tedious,  however,  and  its  discussion  lies  outside  the  scope  of 
this  work. 

(The  reader  is  referred  to  Watson's  "  Theoretical  Astronomy,"  p.  86.  An 
elementary  geometrical  treatment  of  the  reduction  is  also  given  in  Loomis's 
"  Treatise  on  Astronomy,"  p.  211.) 

513.  Determination  of  the  Period  of  a  Planet.  —  This  can  be  done 
in  two  ways : 

First.  By  observation  of  its  node-passage.  When  the  planet  is 
passing  its  node,  it  is  in  the  plane  of  the  ecliptic,  and  the  earth  being 
also  always  in  that  plane,  the  planet's  latitude,  both  geocentric  and 
heliocentric,  will  be  zero,  no  matter  what  may  be  the  place  of  the  earth 
in  its  orbit.  (At  any  other  point  of  the  planet's  orbit  except  the  node 
its  apparent  latitude  would  not  be  thus  independent  of  the  earth's 
place,  but  would  vary  according  to  its  distance  from  the  earth.)  If, 
then,  we  observe  the  planet  at  two  successive  passages  of  the  same 
node,  the  interval  between  the  moments  when  the  latitude  becomes 
zero  will  be  the  planet's  period,  —  exactly,  if  the  node  is  statiouarj' ; 
very  approximately,  even  if  the  node  is  not  absolutely  stationary,  as 
none  of  the  nodes  actually  are. 

There  are  two  difficulties  with  this  method. 

(a)  In  the  case  of  Uranus  the  period  is  eighty-four  years,  and  in  that  of 
Neptune  164  years  —  too  long  to  wait. 

(6)  Since  the  orbits  all  cross  the  ecliptic  at  a  very  small  angle,  so  that  the 
latitude  remains  near  zero  for  a  number  of  days,  it  is  extremely  difficult  to 
determine  the  precise  minute  and  second  when  it  is  exactly  zero ;  and  slight 
errors  in  the  declinations  observed  will  produce  great  errors  in  the  result. 

514.  Second.   By  the  mean   synodic  period  of  the  planet.     The 
synodic  period  is  the  interval  between  two  successive  oppositions  or 
conjunctions  of  the  planet,  the  opposition  being  the  moment  when 
the  planet's  longitude  differs  from  that  of  the  sun  by  180°. 

This  angle  between  the  planet  and  sun  cannot  well  be  measured  directly, 
but  we  can  make  with  the  meridian  circle  a  series  of  observations  both  of 
the  planet's  right  ascension  and  declination  for  several  days  before  and  after 
the  date  of  opposition,  and  reduce  the  observations  to  latitude  and  longitude. 
The  sun  will  be  observed,  of  course,  at  noon,  and  the  planet  near  midnight ; 
but  from  the  solar  observations  we  can  deduce  the  longitudes  of  the  sun 
corresponding  to  the  exact  moments  when  the  planet  was  observed.  From 
these  we  find  the  difference  of  longitude  between  the  planet  and  the  sun  at 
the  time  of  each  planetary  observation ;  and  finally  from  these  differences 


332 


THE  PLANETS. 


of  longitude,  we  find  the  precise  moment  when  that  difference  was  exactly 
180°,  or  the  moment  of  opposition.  This  can  be  ascertained  within  a  very 
few  seconds  of  time  if  the  observations  are  good. 

Since  the  orbits  are  not  strictly  circular,  the  interval  between  two 
successive  observations  will  not  be  the  mean  synodic  period,  but  only 
an  approximation  to  it  ;  but  when  we  know  it  nearly,  we  can  compare 
oppositions  ma^y  years  apart,  and  by  dividing  the  interval  by  the 
known  number  of  entire  synodic  periods  (which  is  easily  determined 
when  we  know  the  approximate  length  of  a  single  period)  we  get  the 
mean  synodic  period  very  closely,  —  especially  if  the  two  oppositions 
occur  at  about  the  same  time  of  the  year.  Having  the  synodic 
period,  the  true  sidereal  period  at  once  follows  from  the  equation 


_ 
P     E     S 

515.  To  find  the  Distance  of  a  Planet  in  Terms  of  the  Earth's 
Distance.  —  When  we  know  the  planet's  sidereal  period,  this  is  easily 
done  by  means  of  two  observations  of  the  planet's  "elongation" 


FIG.  166. — Determination  of  the  Distance  of  a  Planet  from  the  Son. 

taken  at  an  interval  equal  to  its  periodic  time.     The  "  elongation" 
of  a  planet  is  the  difference  between  its  longitude  and  that  of  the 


TO   FIND   THE  DISTANCE   OF   A   PLANET.  333 

sun,  and  a  series  of  meridian-circle  observations  of  sun  and  planet 
will  furnish  these  differences  of  longitude  for  any  selected  moment 
included  within  the  term  of  observation. 

To  find  the  distance  of  the  planet  Mars,  for  instance,  we  must 
therefore  have  two  observations  separated  by  an  interval  of  687  days. 
Suppose  the  earth  to  have  been  at  A  (Fig.  166)  at  the  moment  of  the 
first  observation.  Then  at  the  time  of  the  second  observation  she 
will  be  at  the  point  (7,  the  angle  ASG  being  that  which  the  earth 
will  describe  in  the  next  431  days,  which  is  the  difference  between 
two  complete  years  (or  730 J  days)  and  the  687-day  interval  between 
the  two  observations. 

The  angles  SCM  and  SAM  are  the  "  elongations"  of  the  planet 
from  the  sun,  and  are  given  directly  by  the  observations.  The  two 
sides  SA  and  SO  are  also  given,  being  the  earth's  distance  from 
the  sun  at  the  dates  of  observation.  Hence  we  can  easily  solve 
the  quadrilateral,  and  find  the  length  of  SM^  as  well  as  the  angle 
ASM. 

This  angle  determines  the  planet's  heliocentric  longitude  at  Af,  since 
we  know  the  direction  of  SA,  the  longitude  of  the  earth  at  the  time  of 
observation. 

The  student  can  follow  out  for  himself  the  process  by  which,  from  two 
elongations  of  Venus,  SA  V  and  SB  V,  observed  at  an  interval  of  225  days, 
the  distance  of  Venus  from  the  sun  (or  SV)  can  be  obtained. 

516.  In  order  that  this  method  may  apply  with  strict   accuracy  it  is 
necessary  that  at  the  moment  of  observation  M  should  be  in  the  same 
plane  as  A,  S,  and  C  ;  that  is,  at  the  node.     If  it  is  not  so,  the  process  will 
give  us,  not  the  true  distance  of  the  planet 

itself  from  the  sun,  but  that  of  the  "pro- 
jection "  of  this  distance  on  the  plane  of 
the  ecliptic ;  i.e.,  the  distance  from  the  sun 
to  the  point  m  (Fig.  167),  where  the  per- 
pendicular from  the  planet  would  strike 
that  plane.  But  when  we  have  determined 
Am  and  the  angle  mAM,  the  planet's  geo- 
centric latitude,  we  easily  compute  Mm ;  and  from  Sm  and  Mm  we  get  the 
true  distance  SM  and  the  heliocentric  latitude  of  the  planet  MSm. 

517.  From  a  series  of  pairs  of  observations  distributed  around 
the  planet's   orbit   it  would   evidently  be  possible  to  work  out  the 
orbit  completely.     It  was  in  this  way  that  Kepler  showed  that  the 
orbits  of  the  planets  are  ellipses,  and  deduced  their  distances  from 


334 


THE  PLANETS. 


the  sun ;  and  his  third,  or  harmonic  law,  was  then  discovered  simply 
by  making  a  comparison  between  the  distances  thus  found  and  the 
corresponding  periods. 

518.     Mean  Distance  of  an  Inferior  Planet  by  Means  of  Observa- 
tions of  its  Greatest  Elongation.  —  By  observing  from  the  earth  the 

greatest  elongation  SEV  (Fig. 
168)  of  one  of  the  inferior  planets, 
its  distance  from  the  sun  can  very 
easily  be  deduced  if  we  regard  the 
orbit  as  a  circle ;  for  the  triangle 
SVEwill  be  right-angled  at  V,  and 
SV=  JSE  x  sin  SEV. 

In  the  case  of  Venus  the  orbit  is 
so   nearly  circular   that   the   method 
FIG.  168.  answers  very  well,  the  greatest  elonga- 

Distance  of  an  Inferior  Planet  determined  by     tion   never   differing    much    from  47°. 

Observations  of  its  Greatest  Elongation.        Mercury's  orbit  is  so  eccentric  that  the 

distance  thus  obtained  from  a  single 

elongation  might  be  very  wide  of  the  true  mean  distance.  Since  the  great- 
est elongation,  SEM,  varies  all  the  way  from  18°  to  28°,  it  would  be  neces- 
sary to  observe  a  great  many  elongations,  and  take  the  average  result. 


519.  Deduction  of  the  Orbit  of  a  Planet  from  Three  Observations. 
—  When  one  has  command  of  a  great  number  of  observations  of  a 
planet  running  back  many  years,  and  can  select  such  as  are  conven- 
ient for  his  purpose,  as  Kepler  could  from  Tycho's  records,  it  is 
comparatively  easy  to  find  the  elements  of  a  planet's  orbit ;  but  when 
a  new  planet  is  discovered,  the  case  is  different.  The  problem  first 
arose  practically  in  1801,  when  Ceres,  the  first  of  the  asteroids,  was 
discovered  by  Piazzi  in  Sicily,  observed  for  a  few  weeks  and  then 
lost  in  the  sun's  rays  at  conjunction,  before  other  astronomers  could 
be  notified  of  the  discovery,  in  those  days  of  slow  communication, 
made  slower  and  more  uncertain  by  war. 

Gauss,  then  a  young  man  at  Gottingen,  attacked  the  problem,  and 
invented  the  method  which,  with  slight  modifications,  is  now  univer- 
sally used  in  such  cases. 

We  do  not  propose  to  enter  into  details,  but  simply  say  that  three  ab- 
solutely accurate  observations  of  a  planet's  right  ascension  and  declination 
are  ordinarily  sufficient  to  determine  its  orbit.  Three  observations, 
made  only  as  accurately  as  is  now  possible,  with,  intervals  of  two  or 


PLANETARY  PERTURBATIONS.  335 

three  weeks  between  them,  will  give  a  very  gqod  approximation  to 
the  orbit ;  and  it  can  then  be  corrected  by  further  observations. 

520.  "  Since  there  are  five  independent  variablestin  the  general  equation 
(in  space)  of  a  conic  having  a  given  focus  —  the  sun  —  it  is  necessary  to 
have  five  conditions  in  order  to  determine  them,  "fhree  are  given  by  the 
observations  themselves ;  viz.,  the  directions  of  the  |>ody  as  seen  from  the 
earth  at  three  given  instants;  a  fourth  is  supplied,  by  the  'law  of  equal 
areas,'  since  the  sectors  included  between  the  three  .radii  vectores  must  be 
proportional  to  the  elapsed  times ;  finally,  the  fifth  is  imposed  by  the  require- 
ment that  the  changes  in  the  speed  of  the  body  must  correspond  to  the  vari- 
ations in  the  length  of  the  radius  vector,  in  accordance  with  the  known 
intensity  of  the  sun's  attraction." 

(The  student  is  referred  to  Gauss's  "Theoria  Motus,"  or  to  Watson's 
"Theoretical  Astronomy,"  or  to  Oppolzer's  great  work, on  "The  Determina- 
tion of  Orbits,"  for  the  full  development  of  the  subjectj1,) 

1  •-•  ••*.'•" 


521.     Planetary  Perturbations.  — The  attractio 


each  other  disturbs  their  otherwise  elliptical  motion  around  the  sun. 
As  in  the  case  of  the  lunar  theory  the  disturbing  1  >rces  are,  however, 
always  relatively  small,  but  not  for  the  same  r<ason.  The  sun's 


disturbing  force  is  small  because  its  distance  from 


of  the  planets  for 


he  moon  is  nearly 


four  hundred  times  that  of  the  earth.  In  the  planetary  theory  the 
disturbing  bodies  are  often  nearer  to  the  disturbed  than  is  the  sun 
itself,  as,  for  instance,  in  the  disturbance  of  Saturn  by  Jupiter  at 
certain  points  of  their  orbits  ;  but  the  mass  of  the  disturbing  body  in 
no  case  is  as  great  as  -^^  part  of  the  sun's  mass,  and  for  this  reason 
the  disturbing  force  arising  from  planetary  attraction  is  never  more 
than  a  small  fraction  of  the  sun's  attraction. 

The  greatest  disturbing  force  which  occurs  in  the  planetary  system 
(except  in  the  case  of  some  of  the  asteroids)  is  that  of  Jupiter  on  Saturn 
at  the  time  when  the  planets  are  nearest :  it  then  amounts  to  T^j  of  the 
sun's  attraction.  When  these  two  planets  are  most  remote  from  each  other, 
it  amounts  to  g^7.  There  is  no  other  case  where  the  disturbing  force  is  as 
much  as  j^^  of  the  sun's  attraction  (again  excepting  the  asteroids  disturbed 
by  Jupiter). 

522.  In  any  special  case  the  disturbing  force  can  be  worked  out 
on  precisely  the  same  principles  that  lie  at  the  foundation  of  the 
diagram  by  which  the  sun's  disturbing  force  upon  the  moon  was 
found  (Art.  441,  Fig.  147)  ;  but  the  resulting  diagram  will  look 
very  different,  because  the  disturbing  body  is  relatively  very  neai 
the  disturbed  orbit. 


336  THE   PLANETS. 

The  planetary  perturbations  which  result  from  the  "  integration  " 
of  the  effects  of  the  disturbing  forces,  i.e.,  from  their  continual  action 
through  long  intervals  of  time,  divide  themselves  into  two  great 
classes,  —  the  Periodic  and  the  Secular. 

523.  Periodic  Perturbations.  —  These  are  such  as  depend  on  the 
positions  of  the  planets  in  their  orbits,  and  usually  run  through  their 
course  in  a  few  revolutions  of  the  planets  concerned.     For  the  most 
part  they  are  very  small.     Those  of  Mercury  never  amount  to  more 
than  15",  as  seen  from  the  sun.     Those  of  Venus  may  reach  about 
30",  those  of  the  earth  about  1',  and  those  of  Mars  about  2'.     The 
mutual  disturbances  between  Jupiter  and  Saturn   are  much  larger, 
amounting  respectively  to  28'  and  48' ;  while  those  of  Uranus  are 
again  small,  never  exceeding  3',  and  those  of  Neptune  are  not  more 
than  half  as  great  as  that.     In  the  case  of  the  asteroids,  which  are 
powerfully  disturbed   by   Jupiter,   the   periodical   perturbations   are 
enormous,  sometimes  as  much  as  5°  or  6°. 

524.  Long  Inequalities.  —  The  periodic  inequalities  of  the  planets  are 
so  small,  because,  as  a  rule,  there  is  a  nearly  complete  compensation  effected 
at  every  few  revolutions,  so  that  the  accelerations  balance  the  retardations. 
The  line  of  conjunction  falls  at  random  in  different  parts  of  the  orbits,  and 
when  this  is  the  case,  no  considerable  displacement  of  either  planet  can 
take  place.     But  when  the  periodic  times  of  two  planets  are  nearly  com- 
mensurable, their  line  of  conjunction  will  fall  very  near  the  same  place  in 
the  two  orbits  for  a  considerable  number  of  years,  and  the  small  unbalanced 
disturbance  left  over  at  each  conjunction  will  then  accumulate  in  the  same 
direction  for  a  long  time.     Thus,  five  revolutions  of  Jupiter  roughly  equal 
two  of  Saturn ;  and  still  more  nearly,  seventy-seven  of  Jupiter  equal  thirty- 
one  of  Saturn,  in  a  period  of  913  years.     From  this  comes  the  so-called 
"  long  inequality "  of  Jupiter  and  Saturn,  amounting  to  28'  in  the  place  of 
Jupiter  and  48'  in  that  of  Saturn,  and  requiring  more  than  900  years  to 
complete  its  cycle.     Between  Uranus  and  Neptune  there  is  a  large  inequality 
with  a  period  of  over  4000  years. 

In  the  case  of  the  earth  and  Venus  there  is  a  similar  "  long  inequality  " 
with  a  period  of  235  years,  amounting,  however,  to  less  than  3"  in  the 
positions  of  either  of  the  planets. 

525.  Secular  Inequalities.  —  These  are  inequalities  which  depend 
not  on  the  position  of  the  planets  in  their  orbits,  but  on  the  relative 
position  of  the  orbits  themselves,  with  reference  to  each  other,  —  the 
way,  for  instance,  in  which  the  lines  of  nodes  and  apsides  of  two 
neighboring  orbits  lie  with  reference  to  each  other.     Since  the  plane- 
tary orbits  change  their  positions  very  slowly,  these  perturbations, 


SECULAR   INEQUALITIES.  337 

although  in  the  strict  sense  of  the  word  periodic  also,  are  very  slow 
and  majestic  in  their  march,  and  the  periods  involved  are  such  as 
stagger  the  imagination.  They  are  reckoned  in  n^riads  and  hun- 
dreds of  thousands  of  years.  From  year  to  year  they  are  insignifi- 
cant, but  with  the  lapse  of  time  become  important. 

526.  Secular  Constancy  of  the  Periods  and  Mean  Distances. — 

It  is  a  remarkable  fact,  demonstrated  by  Lagrange  and  La  Place 
about  100  years  ago,  that  the  mean  distances  and  periods  are  en- 
tirely free  from  all  such  secular  disturbance.  They  are  subject  to 
slight  periodic  inequalities  having  periods  of  a  few  years,  or  even 
a  few  hundred  years :  but  in  the  long  run  the  two  elements  never 
change.  They  suffer  no  perturbations  which  depend  on  the  position 
of  the  orbits  themselves,  but  only  such  as  depend  on  the  positions 
of  the  planets  in  their  orbits. 

527.  Revolution  of  the  Nodes  and  Apsides.  —  The  nodes  and  peri- 
helia, on  the  other  hand,  move  on  continuously.     The  lines  of  apsides 
of  all  the  planets  (Venus  alone  excepted)  advance,  and  the  nodes  of 
all  without  exception  (except  possibly  some  of  the  asteroids),  regress 
on  the  ecliptic. 

The  quickest  moving  line  of  apsides  —  that  of  Saturn's  orbit  —  completes 
its  revolution  in  67,000  years,  while  that  of  Neptune  requires  540,000.  The 
swiftest  line  of  nodes  is  that  of  Uranus,  which  completes  its  circuit  in  less 
than  37,000  years,  while  the  slowest  —  that  of  Mercury  —  requires  166,000 
years. 

528.  The  Inclinations  of  the  Orbits.  — These  are  all  slowly  chang- 
ing—  some  increasing,  and  others  decreasing;   but  as  La  Place  and 
Leverrier  have  shown,  all  the  changes  are  confined  within   narrow 
limits  for  all  the  larger  planets  :    they  oscillate   (though  not  in 
regular  periods),  but  the  oscillations  are  never  extensive. 

It  is  not  certain  that  this  is  so  with  the  asteroids,  some  of  which  have 
inclinations  to  the  ecliptic  of  25°  and  30°  :  it  is  possible  that  some  of  these 
inclinations  may  change  by  a  very  considerable  amount. 

529.  The  Eccentricities.  —  These  also  are  slowly  changing  in  the 
.same  way  as  the  inclinations,  some  increasing  and  some  decreasing ; 
and  their  changes  also  are  closely  restricted.     The  periods  of  the 
alternate  increase  and  decrease  are  always  many  thousand  years  in 
length  but,  as  in  the  case  of  the  eccentricities,  they  are  very  irregular  : 
there   is  no  isochronous,  pendulum-like   swing  such  as  many  have 
imagined. 


338  THE   PLANETS. 

The  asteroids  are  again  to  be  excepted ;  the  eccentricities  of  their  orbits 
may  change  considerably. 

530.  Stability  of  the  Planetary  System.  —  About  the  end  of  the 

eighteenth  century  La  Place  and  Lagrange  succeeded  in  proving  that 
the  mutual  attraction  of  the  planets  could  never  destroy  the  system, 
nor  even  change  the  elements  of  the  orbit  of  any  one  of  the  larger 
planets  to  an  extent  which  would  greatly  alter  its  physical  condition. 
The  nodes  and  apsides  revolve  continuously,  it  is  true,  but  that 
change  is  of  no  importance.  The  distances  from  the  sun  and  the 
periods  do  not  change  at  all  in  the  long  run ;  while  the  inclinations 
and  eccentricities,  as  has  just  been  said,  confine  their  variations 
within  narrow  limits. 

531.  The  "Invariable  Plane"  of  the  Solar  System.  —  There  is 
no  reason,  except  the  fact  that  we  live  on  the  earth,  for  taking  the  plane  of 
the  earth's  orbit  (the  plane  of  the  ecliptic)  as  the  fundamental  plane  of  the 
solar  system.     There  is,  however,  in  the  system  an  "invariable  plane"  the 
position  of  which  remains  forever  unchanged  by  any  mutual  action  among 
the  planets,  as  was  discovered  by  La  Place  in  1784.     This  plane  is  denned 
by  the  following  conditions,  —  that  if  from  all  the  planets  perpendiculars  be 
drawn  to  it  (i.e.,  to  speak  technically,  if  the  planets  be  "  projected  "  upon  it), 
and  then  if  we  multiply  each  planet's  mass  by  the  area  which  the  planet's  pro- 
jected radius  vector  describes  upon  this  plane  in  a  unit  of  time,  the  sum  of  these 
products   will  be  a  maximum.     The   ecliptic  is   inclined  about     2°  to  this 
invariable  plane,  and  has  its  ascending  node  nearly  in  longitude  286°. 

532.  La  Place's  Equations  for  the  Inclinations  and  Eccentrici- 
ties. —  La  Place  demonstrated  the  two  following  equations,  viz. : 

(1)  2  (m  Va  X  e2)  =  C.  (2)  5  (m  V^  X  tan20  =  C". 

Equation  (1)  may  be  thus  translated:  Multiply  the  mass  of  each  planet  by  the 
square  root  of  the  semi-major  axis  of  its  orbit,  and  by  the  square  of  its  eccen- 
tricity;  add  these  products  for  all  the  planets,  and  the  sum  will  be  a  constant 
quantity  C,  which  is  very  small.  It  fellows"  that  no  eccentricity  can  become 
very  large,  since  e2  in  the  equation  is  essentially  positive :  there  can  therefore 
be  no  counterbalancing  of  positive  and  negative  eccentricities;  and  if  the 
eccentricity  of  one  planet  increases,  that  of  some  other  planet  or  planets 
must  correspondingly  decrease. 

The  second  equation  is  the  same,  merely  substituting  tan2i  for  e2, »  being 
the  inclination  of  the  planet's  orbit  to  the  invariable  plane. 

The  constant  in  this  case  also  is  small,  though  of  course  not  the  same 
as  in  the  preceding  equation. 


ZA  PLACE'S  EQUATIONS.  339 

533.  Work  of  Poincar&  —  This  has  recently  given  a  new  aspect 
to  the  question  of  the  stability  of  the  system.  Poincare  has  shown 
that  the  assumptions  as  to  the  convergence  of  the  series  used  in  pre- 
vious calculations  are  unwarranted,  and  that  therefore  the  conclu- 
sions reached  are  unsound.  It  is  no  longer  absolutely  certain  that 
gravitational  perturbations  may  not  ultimately  prove  destructive  in 
some  remote  future.  There  are  also  other  conceivable  destructive 
forces,  —  the  action  of  a  resisting  medium,  for  instance,  or  the 
entrance  into  the  system  of  great  bodies  from  outer  space. 


EXERCISES  ON  CHAPTER  XIV. 

1.  What  is  the  mean  daily  gain  of  the  earth  on  Mars  as  seen  from  the 
sun,  i.e.,  the  synodic  motion  of  Mars,  assuming  their  sidereal  periods  as 
365.25  days  for  the  earth,  and  687  days  for  Mars. 

2.  Find  the  synodic  period  of  Venus,  her  sidereal  period  being  225  days. 
(See  Art.  490.) 

3.  Given  the  synodic  period  of  a  planet  as  three  years,  what  is  its 
sidereal  period  ?  A       $  Three-quarters  of  a  year,  or 

(  One  and  a  half  years. 

4.  Given  a  synodic  period  of  four  years,  find  the  sidereal  period. 

5.  What  would  be  the  sidereal  period  of  a  planet  which  had  its  synodic 
period  equal  to  the  sidereal?  Ans.    Two  years. 

6.  Within  what  limits  of  distance  from  the  sun  must  lie  all  planets 
having  synodic  periods  longer  than  two  years?     (Apply  Kepler's  third  law 
after  finding  the  sidereal  periods  that  would  give  a  synodic  period  of  two 
years.)  t  Q.763  Astron.  units,  or    70  895000  miles,  and 

( 1.588       «  »       «  147  500000  miles. 

7.  A  brilliant  starlike  object  was  seen  about  7  P.  M.on  April  1  exactly  at 
the  east  point  of  the  horizon.     Could  it  have  been  a  real  star  or  one  of  the 
planets  ?     If  not,  why  not  ? 

8.  Mercury  was  at  inferior  conjunction  on  Feb.  8,  1896,  at  1  P.M.     On 
May  6,  at  fifteen  minutes  after  noon  (exactly  one  sidereal  period  later), 
its  elongation  from  the  sun  was  observed  to  be  18°  50'  East.     Find  the 
distance  of  the  planet  from  the  sun  at  that  time  in  Astron.  units,  the  earth's 
orbit  being  regarded  as  circular.     (See  Art.  515.) 

The  fact  that  the  first  observation  was  made  at  conjunction  greatly  simplifies  the  cal- 
culation. 

.       (  Distance  from  the  sun  =  0.335  Astron.  units. 

'     The  planet  was  near  perihelion. 


340  THE    PLANETS. 


CHAPTER   XV. 

THE  PLANETS:  METHODS  OF  FINDING  THEIR  DIAMETERS, 
MASSES,  ETC. THE  "  TERRESTRIAL  PLANETS  "  AND  ASTER- 
OIDS. —  INTRA-MERCURIAL  PLANETS  AND  THE  ZODIACAL 
LIGHT. 

IN  discussing  the  individual  peculiarities  of  the  planets,  we  have 
to  consider  a  multitude  of  different  data  ;  for  instance,  their  diameters, 
their  masses,  and  densities,  their  axial  rotation,  their  surface-markings, 
their  reflecting  power  or  "albedo,"  and  their  satellite  systems. 

534,  Diameter.  —  The  apparent  diameter  of  a  planet  is  ascertained 
by  measurement  with  some  kind  of  micrometer  (Art.  73).  For  this 
purpose  the  "double-image"  micrometer  has  an  advantage  over  the 
wire  micrometer  because  of  the  effect  of  irradiation,  and  by  the  fact 
that  in  measuring,  the  observer's  attention  is  concentrated  upon  a 
single  point  instead  of  being  directed  to  two. 

When  we  bring  two  wires  to  touch  the  two  limbs  of  the  planet,  Fig.  169, 

a,  the  bright  image  of  the  planet 

ft is  always  measured  too  large,  be- 

('         }{          \     cause  every  bright  object  appears 
\  A.  )    somewhat  extended  by  its  physi- 

V_X    \^_-^     ological  action  upon  the  retina  of 
the  eye.    This  is  known  as  irradi- 
Micrometer  Measures  of  a  Planet's  Diameter.        ation  —  well    exemplified    at    the 

time  of  new  moon,  when  the  bright 

crescent  appears  to  be  much  larger  than  the  "  old  moon  "  faintly  visible  by 
earth-shine.  With  small  instruments  this  error  is  often  considerable,  but  it 
may  be  reduced  to  some  extent  by  using  a  sufficiently  bright  illumination  of 
the  field  of  view. 

With  the  double-image  micrometer,  the  observer  in  measuring  has  to 
bring  in  contact  two  discs  of  equal  brightness,  as  in  Fig.  169,  b  ;  and  in  this 
case  the  irradiation  almost  vanishes  at  the  point  of  contact. 

The  diameter  thus  measured  is,  of  course,  only  the  apparent  diam- 
eter, to  be  expressed  in  seconds  of  arc,  and  varies  with  every  change 
of  distance.  To  get  the  real  diameter  in  linear  units,  we  have 

AXZ>" 
Keal  diameter  =  ^6265' 


X       f '•  , 

1 
EXTENT  OF  SURFACE  AND  VOLUME.         341 

in  which  A  is  the  distance  of  the  planet  from  the  earth,  and  D"  the 
diameter  in  seconds  of  arc.  If  A  is  given  only  in  astronomical  units, 
the  diameter  comes  out,  of  course,  in  terms  of  that  unit.  To  get 
the  diameter  in  miles,  we  must  multiply  by  the  value  of  this  unit  in 
miles  ;  that  is,  by  the  sun's  distance  from  the  earth. 

535.     Extent  of  Surf  ace  and  Volume.  —  Having  the  diameter,  the 
surface,  of  course,  is  proportional  to  its  square,  and  is  equal  to  the 

earth's  surface  multiplied  by  (  - ),  in  which  s  is  the  semi-diameter  of 


the  planet  and  p  that  of  the  earth. 

/s\3 

The  volume  equals  (  -  )  in  terms  of  the  earth's  volume.     (The  stu- 


dent must  be  on  his  guard  against  confounding  the  volume  or  bulk  of 
a  planet  with  its  mass.) 

The  nearer  the  planet,  other  things  being  equal,  the  more  accurately 
the  above  data  can  be  determined.  The  error  of  O'M  in  measuring 
the  apparent  diameter  of  Venus,  when  nearest,  counts  for  less  than 
thirteen  miles  in  the  real  diameter  of  the  planet ;  while  in  Neptune's 
case  it  would  correspond  to  more  than  1300  miles.  The  student 
must  not  be  surprised,  therefore,  at  finding  considerable  discrepan- 
cies in  the  data  given  for  the  remoter  planets  by  different  authorities. 

536.  Mass  of  a  Planet  which  has  a  Satellite. — In  this  case  its 
mass  is  easily  and  accurately  found  by  observing  the  period  and 
distance  of  the  satellite.  We  have  the  fundamental  equation 


(Jf+m)  = 


in  which  M  is  the  mass  of  the  planet,  m  that  of  its  satellite,  r  the 
radius  of  the  orbit  of  the  satellite,  and  t  its  period. 

The  formula  is  derived  as  follows  :  From  the  law  of  gravitation  the  accel- 
erating force  which  acts  on  the  satellite  is  given  by  the  equation 


f=  r  m 

(Art.  417),  in  which  M  is  the  mass  of  the  planet  and  m  that  of  the  satellite, 
From  the  law  of  circular  motion  (Art.  411,  Eq.  6)  we  have 


342  THE   PLANETS. 

whence  (equating  the  two  values  of  /)  we  have 

M+m 

r2 
and  finally 

(M+m)  = 


This  demonstration  is  strictly  good  only  for  circular  orbits  ;  but  the  equation 
is  equally  true,  and  can  be  proved,  for  elliptical  orbits,  if  for  r  we  put  a,  the 
semi-major  axis  of  the  satellite's  orbit. 

For  many  purposes  a  proportion  is  more  convenient  than  this  equa- 
tion, since  the  equation  requires  that  M,  m,  r,  and  t  be  expressed  in 
properly  chosen  units  in  order  that  it  may  be  numerically  true.  Con- 
verting the  equation  into  a  proportion,  we  have 


, 

or,  in  words,  the  united  mass  of  a  body  and  its  satellite  is  to  the  united 
mass  of  a  second  body  and  its  satellite  as  the  cube  of  the  distance  of 
the  first  satellite  divided  by  the  square  of  its  period  is  to  the  cube  of  the 
distance  of  the  second  satellite  divided  by  the  square  of  its  period.  This 
enables  us  at  once  to  compare  the  masses  of  any  two  bodies  which 
have  attendants  revolving  around  them. 

The  mass  of  the  moon  is  so  considerable  as  compared  with  that 
of  the  earth  (about  ¥^)  that  it  will  not  do  to  neglect  it  ;  but  in  all 
other  cases  the  satellite  is  less  than  -3-^  of  the  mass  of  its  primary, 
and  need  not  be  taken  into  account. 

537.  Examples.  —  (1)  Required  the  mass  of  the  sun  compared  with 
that  of  the  earth.  The  proportion  is 


The  quantities  in  the  last  term  of  the  proportion  are  of  course  the  distance 
and  period  of  the  moon  ;  and  it  is  to  be  remembered  that  for  the  period  of 
the  moon  we  must  use,  not  the  actual  sidereal  period,  but  the  period  as  it 
would  be  if  the  moon's  motion  were  undisturbed,  —  a  period  about  an  hour 
shorter. 

(2)   Compare  the  mass  of  the  earth  with  that  of  Jupiter,  whose  fourth 
satellite  has  a  period  of  16f  days,  and  a  distance  of  1,167,000  miles.  We  have 


(tf  + moon)  :</+ satellite)  = 


MASS    OF    A   PLANET    WHICH    HAS    NO    SATELLITE.        343 

which  gives  the  mass  of  Jupiter  about  316  times  as  great  as  that  of  the  earth 
and  moon  together,  or  318  times  the  mass  of  the  earth  alone. 

538.     It  is  customary  to  express  the  mass  of  a  planet  as  a  certain 
fraction  of  the  sun's  mass,  and  the  proportion  is  simply 

Sun:  Planet  =     -*: 


whence  Planet's  mass  =  Sun's  mass  x  [  — 


where  T  and  R  are  the  planet's  period  and  distance  from  the  sun.  Since 
R  and  r  can  both  be  determined  in  astronomical  units  without  any 
necessity  for  knowing  the  length  of  that  unit  in  miles,  the  masses  oj 
the  planets  in  terms  of  the  sun's  mass  are  independent  of  any  knowl- 
edge of  the  solar  parallax.  But  to  compare  them  with  the  earth,  we 
must  know  this  parallax,  since  the  moon's  distance  from  the  earth, 
which  enters  into  the  equations,  is  found  by  observation  in  miles  or 
in  radii  of  the  earth,  and  not  in  astronomical  units. 

In  order  to  make  use  of  the  satellites  for  this  purpose  we  must 
determine  by  micrometrical  observations  their  distances  from  the 
planets  and  their  periods. 

539.  Mass  of  a  Planet  which  has  no  Satellite.  —  When  a  planet 
has  not  a  satellite,  the  determination  of  its  mass  is  a  very  difficult  and 
troublesome  problem,  and  can  be  solved  only  by  finding  some  pertur- 
bation produced  by  the  planet,  and  then  ascertaining,  by  a  sort  of 
"  trial  and  error"  method,  the  mass  which  would  produce  that  pertur- 
bation.    Venus  disturbs  the  earth  and  Mercury,  and  from  these  per- 
turbations her  mass  is  ascertained.     Mercury  disturbs  Venus,  and 
also  one  or  two  comets  which  come  near  him,  and  in  this  way  we  get 
a  rather  rough  determination  of  his  masSc 

540.  Density.  —  The  density  of   a  body  as   compared  with  the 
earth  is  determined  simply  by  dividing  its  mass  by  its  volume  ;  i.e., 

Density  =  -^ 


•  For  example,  Jupiter's  diameter  is  about  eleven  times  that  of  the  earth 
(i.e.  I  -  \=  11),  so  that  his  volume  is  II8,  or  1331  times  the  earth's.  His  mass, 
derived  from  satellite  observations,  is  about  318  times  the  earth's.  The 


344  THE   PLANETS. 


density,  therefore,  equals  T\\6T'  or  a^out  0.24,  of  the  earth's  density,  or  about 
1£  times  that  of  water,  the  earth's  density  being  5.58  (Art.  171). 

541.  The  Surface  Gravity,  —  The  force  of  gravity  on  a  planet's 
surface  as  compared  with  that  on  the  surface  of  the  earth  is  important 
in  giving  us  an  idea  of  its  physical  condition.     If  r  is  the  radius  of 
the  planet  in  terms  of  the  earth's  radius,  then 

ra  m         f  s\ 

Surface  gravity,  or  y,  =  -r^a»  =  y^-t  X  (  -  J> 

i.e.,it  equals  the  planet's  density,  multiplied  by  its  diameter  expressed 
in  terms  of  the  earth's  diameter. 

Q  1  Q 

For  Jupiter,  therefore,  y  =  -_  =  11  x  density  =  11  x  0.24  =  2.64  nearly. 

That  is,  a  body  at  Jupiter  would  weigh  2.6  times  as  much  as  at  the  earth's 
surface. 

542.  The  Planet's  Oblateness,  —  The    "oblateness"   or    "polar- 
compression"  is  the  difference  between  the  equatorial  and   polar 
diameters  divided  by  the  equatorial  diameter.      It  is,  of  course, 
determined,  when  it  is  possible  to  determine  it  at  all,  simply  by 
micrometric  measurements  of  the  difference  between  the  greatest 
and  least  diameters.     The  quantity  is  always  very  small  and  the 
observations  delicate. 

543.  The  Time  of  Rotation,  when  it  can  be  determined,  is  found 
by  observing  the  passage  of  some  spot  visible  in  the  telescope  across 
the  central  line  of  the  planet's  disc.     In  reducing  the  observations 
they  must  be  corrected  for  changes  in  the  planet's  direction  from  the 
earth,  and  also  for  variations  of  distance  which  affect  the  time  in 
which  light  reaches  us.    In  some  cases  the  rotation  period  has  been  de- 
termined by  observation  of  regular  changes  in  the  planet's  brightness. 

544.  The  Inclination  of  the  Axis  is  deduced  from  the  same  obser. 
vations  which  are  used  in  obtaining  the  rotation-period.    It  is  neces- 
sary to  determine  with  the  micrometer  the  paths  described  by  differ- 
ent spots  as  they  move  across  the  planet's  disc.     It  is  possible  to 
ascertain  it  with  accuracy  for  only  a  very  few  of  the  planets  :  Mars, 
Jupiter,  and  Saturn  are  the  only  ones  that  furnish  the  needed  data. 

545.  The  Surface  Peculiarities  and  Topography  of  the  surface 
are  studied  by  the  telescope.     The  observer  makes  drawings  of  any 


SPECTROSCOPIC   PECULIARITIES   AND  ALBEDO.  345 

markings  which  he  may  see,  and  by  their  comparison  is  at  last  able 
to  discriminate  between  what  is  temporary  and  what  is  permanent  on 
the  planet.1  Mars  alone,  thus  far,  permits  us  to  make  a  map  of  its 
surface. 

546.  Spectroscopic    Peculiarities   and  Albedo. — The   character- 
istics of  the  planet's  atmosphere  can  be  to  some  extent  studied  by 
means  of  the  spectroscope,  which  in  some  few  cases  shows  the  pres- 
ence of  water- vapor  and  other  absorbing  media,  by  dark  bands  in 
the   planet's    spectrum.     The    u  albedo ,"  or   reflecting   power   of    a 
planet's  surface   is    determined   by  photometric   observations,  com- 
paring it  with  a  real  or  artificial  star,  or  with  some  other  planet. 

547.  The  Satellite  System  of  a  Planet.  —  The  principal  data  to  be 
ascertained  are  the  distances  and  periods  of  the  satellites,  and  the 
observations   are   made  by   measuring  the   apparent   distances   and 
directions  of  the  satellites  from  the  centre  of  the  planet  with  the  wire 
micrometer  (Art.   73).     Observations   made  at  the  times  when  the 
satellite  is  near  its  elongation  are  especially  valuable  in  determining 
the  distance. 

If  the  planet  and  earth  were  at  rest,  the  satellite's  path  would  appear  to 
be  an  ellipse,  unaltered  in  dimensions  during  the  whole  series  of  observa- 
tions ;  but  since  the  earth  and  planet  are  both  moving,  it  becomes  a  compli- 
cated problem  to  determine  the  satellite's  true  orbit  from  the  ensemble  of 
observations. 

548.  With  the  exception  of  the  moon  and  the  eighth  satellite  of  Saturn, 
most  of  the  satellites  of  the  planetary  system  move  nearly  in  the  plane  of 
the  equator  of  the  primary ;  and  all  but  the  moon  and  the  outer  satellites 
of  Jupiter  and  Saturn  move  in  orbits  almost  circular.     La  Place  has  shown 
that  if  satellites  originally  moved  in  orbits  nearly  coincident  with  the  plane 
of  the  planet's  equator,  its  equatorial  protuberance  would  tend  to  retain 
them  in  that  plane,  but  the  almost  perfect  circularity  of  the  orbits  is  not 
yet  explained.     When  there  are  a  number  of  satellites  in  a  system,  inter- 
esting problems  arise  in  connection  with  their  mutual  disturbances;  and 
in  a  few  cases  it  becomes  possible  to  determine  a  satellite's  mass  as  com- 
pared with  that  of  its  primary.     In  several  instances  satellites  show  pecul- 
iar variations  in  their  brightness,  which  are  supposed  to  indicate  that  they 
make  an  axial  rotation  in  the  time  of  one  revolution  around  the  primary, 
in  the  same  way  as  our  moon  does. 

1  Photography  is  beginning  to  be  applied,  and  with  some  success,  in  the  case 
of  Mars  and  Jupiter. 


346  THE    PLANETS. 

549.  Humboldt's  Classification  of  the  Planets,  —  Humboldt  has 
divided  the  planets  into  two  groups :    the  terrestrial  planets,   so- 
called,  and  the  major  planets.     The  terrestrial  planets  are  Mercury, 
Venus,  the  earth,  and  Mars.     They  are  bodies  of  the  same  order  of 
magnitude,  ranging  from  3000  to  8000  miles  in  diameter,  not  very 
different  in  density  (the  earth  being  the  largest  and  probably  the 
densest  of  them),  and  are  probably  roughly  alike  in  physical  consti- 
tution, and  covered  with  water  and  air.     But  we  hasten  to  say  that 
the  differences  in  the  amount  of  heat  and  light  which  they  receive 
from  the  sun,  and  in  the  force  of  gravity  upon  their  surfaces,  and 
probably  in  the  density  of  their  atmospheres,  are  such  as  to  bar  any 
positive  conclusions  as  to  their  being  the  abode  of  life  resembling  the 
forms  of  life  with  which  we  are  acquainted  on  the  earth. 

550.  The  four  major  planets,  Jupiter,  Saturn,  Uranus,  and  Nep- 
tune, are  much  larger  bodies  (ranging  in  diameter  between  30,000 
and  90,000  miles),  are  much  less  dense,  and  so  far  as  we  can  make 


FIG.  170.  —Relative  Sizes  of  the  Planets. 

out,  present  to  us  only  a  surface  of  cloud,  and  may  not  have 
anything  solid  about  them.  There  are  some  reasons  for  suspecting 
that  they  are  at  a  high  temperature  ;  in  fact,  that  Jupiter  is  a  sort  of 
semi-sun;  but  this  is  by  no  means  yet  certain. 

As  for  the  multitudinous  asteroids,  the  probability  is  that  they 
represent  a  single  planet  of  the  terrestrial  group  which,  as  has 
been  intimated,  failed  for  some  reason  in  its  evolution,  or  else  has 


MERCURY.  347 

been  broken  to  pieces.  All  of  them  united  would  not  make  a  planet 
one-hundredth  the  mass  of  the  earth. 

Fig.  170  shows  the  relative  sizes  of   the  different  planets. 

In  what  follows,  all  the  numerical  data,  so  far  as  they  depend  on 
the  solar  parallax,  are  determined  on  the  assumption  that  that  paral- 
lax is  8". 80,  and  that  the  sun's  mean  distance  is  92,897000  miles. 

MERCURY. 

551 .  There  is  no  record  of  the  discovery  of  the  planet.     It  has 
been  known  from  remote  antiquity ;  and  we  have  recorded  observa- 
tions running  back  to  B.C.  264. 

For  a  time  the  ancient  astronomers  seem  to  have  failed  to  recognize  it 
as  the  same  body  on  the  eastern  and  western  sides  of  the  sun,  so  that  the 
Greeks  had  for  a  time  two  names  for  it,  —  Apollo  when  it  was  morning  star, 
and  Mercury  when  it  was  evening  star.  According  to  Arago,  the  Egyptians 
called  it  Set  and  Horus,  and  the  Hindoos  also  gave  it  two  names. 

It  is  so  near  the  sun  that  it  is  comparatively  seldom  seen  with  the 
naked  eye  ;  but  when  near  its  greatest  elongation  it  is  easily  enough 
visible  as  a  brilliant  star  of  the  first  magnitude  low  down  in  the  twi- 
light, perhaps  not  quite  so  bright  as  Sirius,  but  certainly  brighter  than 
Arcturus.  It  is  usually  visible  for  about  a  fortnight  at  each  elonga- 
tion, and  is  best  seen  in  the  evening  at  such  eastern  elongations  as 
occur  in  March  and  April.  In  Northern  Europe  it  is  much  more 
difficult  to  observe  than  in  lower  latitudes,  and  Copernicus  is  said 
never  to  have  seen  it.  Tvcho,  however,  obtained  a  considerable 
number  of  observations.  For  the  most  part,  of  course,  observations 
upon  it  are  made  in  the  daytime. 

552.  It  is  exceptional  in  the  solar  system  in  a  great  variety  of 
ways.    It  is  the  nearest  planet  to  the  sun;  receives  the  most  light  and 
heat,  is  the  swiftest  in  its  movement,  and  (excepting  some  of  the 
asteroids)  has  the  most  eccentric  orbit,  with  the  greatest  inclination  to 
the  ecliptic.     It  is  also  the  smallest  in  diameter  and  has  the  least 
mass,  asteroids  again  excepted. 

553.  Distance,  Light,  and  Heat.  —  Its  mean  distance  from  the 
sun  is  36,000000  miles,  but  the  eccentricity  of  its  orbit  is  so  great 
(0.205),  that  the  sun  is  seven  and  one-half  millions  of  miles  out  of 
its  centre,  and  the  actual  distance  of  the  planet  from  the  sun  ranges 
all  the  way  from  28,500000  to  43,500000,  while  its  velocity  in  its  orbit 


348  MERCURY. 

varies  from  thirty -five  miles  a  second  at  perihelion  to  only  twenty-three 
at  aphelion.  On  the  average  it  receives  6.7  times  as  much  light  and 
heat  as  the  earth ;  but  the  heat  received  at  perihelion  is  to  that  at 
aphelion  in  the  ratio  of  9  to  4.  For  this  reason  there  must  be  two 
seasons  in  its  year  due  to  the  changing  distance,  even  if  the  equator 
of  the  planet  is  parallel  to  the  plane  of  its  orbit,  which  would  preclude 
seasons  like  our  own.  If  the  planet's  equator  is  inclined  at  an  angle 
like  the  earth's,  then  the  seasons  must  be  very  complicated. 

554.  Period. — The  sidereal  period  is  very  nearly  88  days,  and 
the  synodic  period,  or  the  time  from  conjunction  to  conjunction  again, 
is  about  116  days.     The  greatest  elongation  ranges  from  18°  to  28°, 
and   occurs   about  twenty-two   days   before   and   after  the   inferior 
conjunction,  or  about  thirty-six  days  before  and  after  the  superior 
conjunction.      The  planet's  arc  of  retrogression  is  about  12°  (consid- 
erably variable),  and  the  stationary  point  is  very  near  the  greatest 
elongation. 

555.  Inclination.  —  The  inclination  of  the  orbit  to  the  ecliptic  is 
about  7°,  but  the  greatest  geocentric  latitude  (that  is,  the  planet's 
greatest  distance  from  the  ecliptic  as  seen  from  the  earth)  is  never 
quite  so  great. 

556.  Diameter,  Surface,  and  Volume.  —  The  apparent  diameter 
ranges  from  5"  to  about  13",  according  to  its  distance  from  us  ;  the 
least  distance  from  the  earth  being   about   57,000000   miles  (93  — 
36),  while   the   greatest  is  about  129,000000   (93  +  36).     The  real 
diameter  is  very  near  3000  miles,  not  differing  from  that  more  than 
fifty  miles  either  way.     It  is  not  easy  to  measure,  and  the  "  probable 
error"  is  perhaps   rather   larger  than  would   have   been   expected. 
With    this    diameter,    its    surface    is    |    of    the    earth's,    and    its 
volume  j|g. 

557.  Mass,  Density,  and  Surface  Gravity.  —  Its  mass  is  very  diffi- 
cult to  determine,  since  it  has  no  satellite,  and  the  values  obtained 
by  La  Place,  Encke,  Leverrier,  and  others,  range  all  the  way  from  £  of 
the  earth's  mass  to  -fa.     The  planet  is  so  small  and  so  near  the  sun 
that  its  effect  in  disturbing  the  other  planets  is  very  slight,  and  the 
"  probable  error  "  of  the  mass  determined  from  these  perturbations 
is  correspondingly  large. 


TELESCOPIC   APPEARANCE   AND   PHASES. 


349 


In  his  recent  work  upon  the  "Fundamental  Elements  of  Astronomy," 
Newcomb  settles  upon  a  value  of  ^^ ^^  of  the  sun's  mass,  or  ^T  of  the 
earth's.  Harkness  gets  ^  of  the  earth's.  Assuming  Newcomb's  value,  the 
density  of  Mercury  comes  out  about  seven-eighths  that  of  the  earth ;  and  its 
surface  gravity  a  little  less  than  one-third.  If  we  take  Harkness'  figures  the 
density  is  only  0.72,  and  its  superficial  gravity,  0.27.  But  none  of  the  results 
thus  far  obtained  are  to  be  regarded  as  more  than  rough  approximations  to 
the  truth.  The  data  are  not  sufficient  to  furnish  accurate  determinations. 


558.  Its  Albedo,  or  reflecting  power,  as  determined  by  Zollner  is 
very  low  —  only  0.13,  somewhat  inferior  to  that  of  the  moon. 

In  1878  Mr.  Nasmyth  observed  the  planet  in  the  same  field  of  view  with 
Venus;  and  although  Mercury  was  then  not  much  more  than  half  as  far 
from  the  sun  as  Venus,  and  therefore  four  times  as  brightly  illuminated,  it 
appeared  to  be  less  luminous  in  the  telescope.  "Venus  was  like  silver, 
Mercury  like  zinc  or  lead." 

In  the  proportion  of  light  given  out  at  its  different  phases,  it 
behaves  like  the  moon,  flashing  out  strongly  near  the  full,  as  if  it  had 
a  surface  of  the  same  rough  structure  as  that  of  our  satellite.  Like 
the  moon  and  Mars  also,  but  in  contrast  with  Jupiter,  the  illuminated 
edge  of  its  disc  is  always  brighter  than  the  centre. 

559.  Telescopic  Appearance  and  Phases.  —  Seen  by  the  telescope, 
the  planet  looks  like  a  little  moon,  showing  phases  precisely  similar 
to  those  of  our  satellite.     At  inferior  conjunction  the  dark  side  is 
towards  us;  at  superior  conjunction  the  illuminated  surface.     At 


FIG.  171.  —  Phases  of  Mercury  and  Venus. 


the  greatest  elongation  it  appears  like  a  half-moon.  Between  supe- 
rior conjunction  and  greatest  elongation  it  is  gibbous,  while  between 
inferior  conjunctions  and  the  elongations  it  shows  the  crescent  phase. 
Fig.  171  illustrates  the  phases  of  both  Mercury  and  Venus. 


350  MERCURY. 

As  a  rule  Mercury  is  so  near  the  sun  that  it  can  be  observed  only 
in  the  daytime,  but  with  proper  precautions  in  screening  the  object 
glass  of  the  telescope  from  direct  sunlight,  the  observation  is  not 
very  difficult.  The  surface  presents  very  little  of  interest,  there 
being  no  well-defined  markings,  though  there  are  sometimes  indistinct 
shadings  which  perhaps  indicate  permanent  geographical  features. 
It  has  been  attempted  to  deduce  from  these  the  length  of  the  planet's 
day,  and  many  years  ago  Schroeter,  a  German  astronomer,  a  con- 
temporary of  the  elder  Herschel,  obtained  24h  05m  as  a  result,  which 
until  recently  remained  practically  uncontradicted,  though  also 
unconfirmed.  In  1889,  however,  Schiaparelli,  the  Italian  astronomer, 
announced  that  he  had  ascertained  that  the  markings  do  not  move 
sensibly  upon  the  planet's  disc,  in  the  course  of  several  hours  even, 
and  therefore,  that  the  time  of  rotation  must  be  much  longer  than 
a  day,  and  he  finds  as  a  result  the  remarkable  fact  that  the  planet 
rotates  on  its  axis  only  once  during  its  orbital  period  of  88  days;  and 
keeps  the  same  face  always  turned  towards  the  sun,  behaving  in  this 
respect,  just  as  the  moon  does  towards  the  earth.  Owing  to  the 
eccentricity  of  the  orbit,  however,  the  planet  has  a  large  'libration' 
(Art.  250),  amounting  to  nearly  23-J-0  on  each  side  of  the  mean 
position ;  i.e.,  seen  from  a  favorable  position  on  the  planet's  surface, 
the  sun  instead  of  rising  and  setting  daily,  as  with  us,  would  oscillate 
about  47°  back  and  forth  in  the  sky,  every  88  days.  This  asserted 
discovery  is  extremely  important,  and  has  excited  great  interest. 
There  is  little  doubt  that  it  is  correct ;  but  the  necessary  observa- 
tions are  very  difficult,  and  the  only  direct  confirmation  thus  far  is 
from  the  Flagstaff  observations  of  Lowell  in  1896,  who  gets  a  result 
perfectly  agreeing  with  that  of  Schiaparelli. 

560.  Atmosphere.  —  The  evidence  upon  this  subject  is  not  con- 
clusive. Its  atmosphere,  if  it  has  one,  must,  however,  be  much  less 
dense  than  that  of  Venus.  No  ring  of  light  is  seen  surrounding  the 
disc  of  the  planet  when  it  enters  the  limb  of  the  sun  at  the  time  of 
a  transit,  while  in  the  case  of  Venus  such  a  ring,  due  to  the  atmos- 
pheric refraction,  is  very  conspicuous.  On  the  other  hand,  Huggins 
and  Vogel,  who  have  examined  the  spectrum  of  the  planet,  report 
that  certain  lines  in  the  spectrum,  due  to  the  presence  of  water- 
vapor,  were  decidedly  stronger  than  in  the  spectrum  of  the  air  (illu- 
minated by  sunshine),  which  formed  the  background  for  the  planet, 
making  it  probable  that  it  has  an  atmosphere  containing  water-vapor 
like  the  atmosphere  of  the  earth,  but  probably  less  extensive  and 
dense. 


INTERVAL   BETWEEN   TRANSITS.  351 

561.  Transits.  —  Usually  at  the  time  of  inferior  conjunction  the 
planet  passes  north  or  south  of  the  sun,  the  inclination  of  its  orbit 
being  7°;  but  if  the  conjunction  occurs  when  the  planet  is  very  near 
its  node,  it  will  cross  the  disc  of  the  sun  and  be  visible  upon  it  as  a 
small  black  spot  —  not,  however,  large  enough  to  be  seen  without  a 
telescope,  as  Venus  can  under  similar  circumstances. 

At  this  time  we  have  the  best  opportunity  for  measuring  the  diameter  of 
the  planet ;  but  unless  special  precautions  are  taken,  the  measured  diameter 
under  these  circumstances  is  likely  to  be  too  small,  on  account  of  the  irradia- 
tion of  the  surrounding  background,  which  encroaches  upon  the  planet's  disc. 

Since  the  planet's  nodes  are  in  longitudes  227°  and  47°,  and  are 
passed  by  the  earth  on  May  7  and  November  9,  the  transits  can 
occur  only  near  those  days.  If  the  orbit  of  the  planet  were  strictly 
circular,  the  " transit  limit"  (corresponding  to  an  ecliptic  limit) 
would  be  2°  10';  but  at  the  May  transits  the  planet  is  near  its 
aphelion  and  much  nearer  the  earth  than  ordinarily,  so  that  the 
limit  is  diminished,  while  the  November  limit  is  correspondingly 
increased.  The  May  transits  are  in  fact  less  than  half  as  numerous 
as  the  November  transits. 

562.  Interval  between  Transits.  —  Twenty-two  synodic  periods 
of  Mercury  are  pretty  nearly  equal  to  7  years ;  41  still  more  nearly 
equal  13  years;   and  145  almost  exactly  equal  46  years.     Hence, 
after  a  November  transit,  a  second  one  is  possible  in  7  years,  prob- 
able in  13  years,  and  practically  certain  in  46.     For  the  May  tran- 
sits the  repetition  after  7  years  is  not  possible,  and  it  often  fails  in 
13  years. 

The  first  transit  of  Mercury  ever  observed  was  by  Gassendi,  Nov.  7, 1631. 

The  last  transit  (visible  in  the  U.  S.)  occurred  on  Nov.  14,  1907. 

The  following  list  gives  the  transits  of  the  present  century.  An  asterisk 
denotes  that  the  whole  transit  will  be  visible  in  the  U.  S. ;  a  dagger,  that  a 
part  of  it  can  be  seen. 

f!907,  Nov.  14 j  f!914,  Nov.    6;     1924,  May    7; 

1927,  Nov.  8;  1937,  May  10;  1940,  Nov.  12; 

*1953,  Nov.  13;  *1960,  Nov.  6;  f!970,  May  9; 

f!973,  Nov.  9;  1986,  Nov.  12;  1999,  Nov.  14. 

The  transits  of  Mercury  are  of  no  particular  astronomical  importance, 
except  as  giving  accurate  determinations  of  the  planet's  place,  by  means  of 
which  its  orbit  can  be  determined.  Newcomb  has  also  recently  made  an 
investigation  of  all  the  recorded  transits,  for  the  purpose  of  testing  the 
uniformity  of  the  earth's  rotation.  They  seem  to  indicate  certain  small 
irregularities  in  that  motion,  but  hardly  make  the  fact  certain. 


352  VENUS. 


VENUS. 

563.  The  next  planet  in  order  from  the  sun  is  Venus,  the  brightest 
and  most  conspicuous  of  all ;  the  earth's  twin  sister  in  magnitude, 
density,  and  general  constitution,  if  not  also  in  age,  as  to  which  we 
have  no  knowledge.     Like  Mercury,  it  had  two  names  among  the 
Greeks,  —  Phosphorus  as  morning  star,  and  Hesperus  as  evening  star. 
It  is  so  brilliant  that  it  is  easily  seen  by  the  naked  eye  in  the  day- 
time for  several  weeks  when  near  its  greatest  elongation  ;  sometimes 
it  is  bright  enough  to  catch  the  eye  at  once,  but  usually  it  is  seen  by 
daylight  only  when  one  knows  precisely  where  to  look  for  it. 

(There  is  no  good  reason  to  suppose  that  it  is  the  "Star  of  Bethlehem," 
though  some  have  imagined  this  to  be  the  case.) 

564.  Distance,  Period,  and  Inclination  of  Orbit.  —  Its  mean  dis- 
tance from  the  sun  is  67200000  miles.     The  eccentricity  of  the 
orbit  is  the  smallest  in  the  planetary  system  (only  0.007),  so  that 
the  greatest  and  least  distances  of  the  planet  from  the  sun  differ 
from  the  mean  only  470000  miles  each  way.     Its  orbital  velocity  is 
twenty-two  miles  per  second. 

Its  sidereal  period  is  225  days,  or  seven  and  one-half  months,  and 
its  synodic  period  584  days  —  a  year  and  seven  months.  From  supe- 
rior conjunction  to  elongation  on  either  side  is  220  days,  while  from 
inferior  conjunction  to  elongation  is  only  71  or  72  days.  The  arc 
of  retrogression  is  16°. 

The  inclination  of  its  orbit  is  only  3^-°. 

565.  Diameter,  Surface,  and  Volume.  —  The  apparent  diameter 
ranges  from  67"  at  the  time  of  inferior  conjunction  to  only  11"  at  the 
superior.     This  great  difference  depends,  of  course,  upon  the  enor- 
mous change  in  the  distance  of  the  planet  from  the  earth.     At 
inferior  conjunction  the  planet  is  only- 26  000000  miles  from  us  (93 
—  67).     No  other  body  ever  comes  so  near  the  earth  except  the 
moon,  an  occasional  comet,  and  Eros.1    Its  greatest  distance  at  supe- 
rior conjunction  is  160  000000  miles  (93  +  67),  so  that  the  ratio 
between  the  greatest  distance  and  the  least  is  more  than  6  to  1. 

The  real  diameter  of  the  planet  is  7700  (±  30)  miles.  Its  sur- 
face, as  compared  with  that  of  the  earth,  is  ninety-five  per  cent ;  its 
volume  ninety-two  per  cent. 

1  See  page  377  for  note  on  Eros. 


MASS,    DENSITY,    AND    GRAVITY.  353 

566.  Mass,  Density,  and  Gravity.  —  By  means  of  the  perturba- 
tions she  produces,  the  mass  of  Venus  is  found,  according  to  New- 
comb,  to  be  about  eighty-two  per  cent  of  the  earth's ;  hence  her 
density  is  eighty-eight  per  cent,  and  her  superficial  gravity  eighty-five 
per  cent  of  the  earth's. 

567.  Phases.  —  The  telescopic  appearance  of  the  planet  is  strik- 
ing on  account  of  her  great  brilliance.    When  about  midway  between 
greatest  elongation  and  inferior  conjunction  she  has  an  apparent 
diameter  of  40",  so  that,  with  a  magnifying  power  of  only  forty-five, 
she  looks  exactly  like  the  moon  four  days  old,  and  of  precisely  the 
same  apparent  size. 


FIG.  172.  —Telescopic  Appearances  of  Venus. 

Very  few  persons,  however,  would  think  so  on  their  first  view  through 
the  telescope,  for  a  novice  always  underrates  the  apparent  size  of  a  tele- 
scopic object :  he  instinctively  adjusts  his  focus  as  if  looking  at  a  picture 
only  a  few  inches  away,  instead  of  projecting  the  object  visually  into 
the  sky. 

According  to  the  theory  of  Ptolemy,  Venus  could  never  show  us 
more  than  half  her  illuminated  surface,  since  according  to  his  hypoth- 
esis she  was  always  between  us  and  the  supposed  orbit  of  the  sun 
(Art.  500).  Accordingly,  when  in  1610  Galileo  discovered  that  she 
exhibited  the  gibbous  phase  as  well  as  the  crescent,  it  was  a  strong 
argument  for  Copernicus.  Galileo  announced  his  discovery  in  a 
curious  way,  by  publishing  the  anagram,  - — 

"Haec  immatura  a  me  jam  frustra  leguntur ;  o.  y." 


354  VENUS. 

Some  months  later  he  furnished  the  translation,  — 
"  Cynthiae  figuras  aemulatur  Mater  Amorum," 

which  is  formed  by  merely  transposing  the  letters  of  the  anagram. 
His  object  was  to  prevent  any  one  from  claiming  to  have  anticipated 
him  in  this  discovery,  as  had  been  done  with  respect  to  his  discovery 
of  the  sun  spots. 

Fig.  172  represents  the  disc  of  the  planet  as  seen  at  four  points  in  its 
orbit.  1,  3,  and  5  are  taken  at  superior  conjunction,  greatest  elongation, 
and  near  inferior  conjunction  respectively,  while  2  and  4  are  at  interme- 
diate points. 

568.  Maximum  Brightness.  —  The  planet  attains  its  maximum 
brilliance  thirty-six  days  before  and  after  inferior  conjunction,  at  a 
distance  of  about  38°  or  39°  from  the  sun,  when  its  phase  is  like  that 
of  the  moon  about  five  days  old.     It  then  casts  a  strong  shadow, 
and,  as  has  already  been  said,  is  easily  visible  by  day  with  the 
naked  eye. 

569.  Surface  Markings.  —  These  are  not  at  all  conspicuous.    Near 
the  limb  of  the  planet,  which  is  always  much  brighter  than  the  central  parts 
(as  is  also  the  case  with  Mercury  and  Mars),  they  can  never  be  well  seen, 
although  sometimes  when  Venus  was  in  the  crescent  phase,. intensely  bright 
spots  have  been  reported  near  the  cusps,  as  at  a  and  b  in  No.  4,  Fig.  172. 
These  may  perhaps  be  ice-caps  like  those  which  are  seen  on  Mars.    Near  the 
"  terminator,"  which  is  less  brilliant  and  less  sharply  defined  than  the  limb, 
irregular  darkish  shadings  are  sometimes  seen,  such  as  are  indicated  by  the 
dotted  lines  in  the  figures,  but  without  any  distinct  outline.     They  may 
be  continents  and  oceans  dimly  visible,  or  they  may  be  mere  atmospheric 
objects ;  observations  do  not  yet  decide. 

Mr.  Lowell,  however,  until  recently  alone  among  observers,  describes  a 
very  different  aspect  according  to  his  Flagstaff  studies.  He  makes  out  an 
obvious  and  permanent  system  of  markings,  consisting  of  rather  narrow 
dark  streaks,  nearly  straight,  radiating  in  a  spoke-like  manner  from  a  sort 
of  "  hub  "  near  the  centre  of  the  planet's  disc.  They  seemed  to  him  to  be 
quite  sharp  in  outline,  but  dim,  as  if  seen  through  a  luminous  atmosphere 
of  considerable  depth.  He  even  goes  so  far  as  to  offer  a  map  of  the  planet, 
with  names  appended  to  some  of  the  principal  features.  He  attributes  his 
success,  not  so  much  to  the  power  of  his  twenty-four-inch  telescope  (one  of 
the  last  and  most  perfect  of  Clark's  productions),  as  to  the  excellence  of  the 
atmospheric  conditions  at  Flagstaff.  He  maintains,  very  reasonably  it 
seems,  that  in  the  observation  of  objects  like  the  finer  markings  on  the  discs 
of  Mercury,  Venus,  and  Mars,  steadiness  of  the  image  is  even  more  impor- 
tant than  telescopic  power  or  acuteness  of  vision.  He  also  notes  that  the 


ROTATION   OF   THE   PLANET.  855 

visibility  of  the  markings  is  in  a  measure  dependent  upon  the  phase,  as 
they  are  seen  most  easily  when  the  disc  is  nearly  full. 

570.  Rotation  of  the  Planet.  —  The  rotation-period  of  the  planet 
is  still  a  subject  of  dispute.     Schroeter,  from  his  observations  of 
shadings  noted  upon  its  surface,  deduced  a  "day"  of  23h  21m,  and 
some  more  recent  observers  support  his  conclusion.     On  the  other 
hand  Schiaparelli,  while  he  does  not  profess  to  have  yet  determined 
the  period  with  precision,  considers  that  his  observations  disprove 
Schroeter's  result,  and  show  that  the  rotation-period  must  be  long, 
and  probably  225  days,  identical  with  the  planet's  orbital  period,  as 
in  the  case  of  Mercury.     Mr.  Lowell's  observations  of  1896  confirm 
this  conclusion,  and  are   indeed  decisive  if  they  are  accepted  as 
correct. 

It  is  not  unlikely  that  the  spectroscope  will  soon  give  us  a 
final  settlement  of  the  question :  if  the  rotation  is  rapid  the  dark 
lines  in  the  spectrum  must  be  displaced  at  the  edges  of  the  planet's 
disc  (Art.  321,  note)  by  an  amount  that  can  be  measured.1 

De  Vico,  fifty  years  ago,  concluded  that  the  planet's  equator  makes 
an  angle  of  54°  with  the  plane  of  its  orbit,  and  the  statement  is  still 
found  in  many  text-books,  though  it  is  probably  incorrect.  If  the 
bright  spots  referred  to  in  Art.  569  are  really  "polar  caps"  the 
inclination  must  be  small. 

No  sensible  difference  has  been  ascertained  between  the  different  diam- 
eters of  the  planet,  a  fact  which  favors  Schiaparelli's  rotation-period.  If  it 
were  really  as  much  flattened  at  the  poles  as  the  earth  is,  there  should  be  a 
measurable  difference  of  0".2  between  the  polar  and  equatorial  diameters. 

571.  Mountains.  —  From   certain   irregularities   occasionally   ob- 
served upon  the  terminator,  and  especially  from  the  peculiar  blunted 
form  of  one  of  the  cusps  of  the  crescent,  various  observers  have  con- 
cluded that  there  are  numerous  high  mountains  upon  the  surface  of 
the  planet.     Schroeter  assigned  to  some  of  those  near  the  southern 
pole  the  extravagant  altitude  of  twenty-five  or  thirty  miles,  but  the 
evidence  is  entirely  insufficient  to  warrant  any  confidence  in  the 
conclusion. 

572.  Albedo.  —  According  to  Zollner  the  Albedo  of  the  planet  is 
0.50,  which  is  about  three  times  that  of  the  moon,  and  almost  four 
times  that  of  Mercury.     It  is,  however,  exceeded  by  the  reflecting 
power  of  the  surfaces  of  Jupiter  and  Uranus,  while  that  of  Saturn 

1  Up  to  1908  the  results  on  the  whole  rather  favor  the  long  period. 


356  VENUS. 

appears  to  be  about  the  same.  This  high  reflecting  power  probably 
indicates  that  the  surface  is  mostly  covered  with  cloud,  as  few  rocks 
or  soils  could  match  it  in  brightness.  Lowell,  however,  denies  the 
existence  of  anything  like  a  nearly  continuous  cloud  veil  such  as 
has  been  generally  supposed. 

573.  Evidences  of  Atmosphere. — When   the   planet  is  near  the 
sun,  the  horns  of  the  crescent  extend  notably  be3*ond  the  diameter, 
and  when  very  near  the  sun,  a  thin  line  of  light  has  been  seen  by 
several   observers,  especially  Professor    Lyman  of   New  Haven,  to 
complete  the  whole  circumference.     This  is  due  to  refraction  of  sun- 
light by  the  planet's  atmosphere,  a  phenomenon  still  better  seen  as 
the   planet   is    entering  upon  the  sun's  disc  at  a  transit,   when    the 
black  disc  is  surrounded  by  a  beautiful  ring  of  light.     From  the  ob- 
servations of  the  transit  of  1874,  Watson  concluded  that  the  planet's 
atmosphere  must  have  a  depth  of  about  fifty-five  miles,  that  of  the 
earth  being  usually  reckoned  at  forty  miles.     Later  observations,1 
however,  indicate  an  atmosphere  less  extensive  than  our  own,  and 
that  the  luminous  twilight  ring  is  due  rather  to  diffuse  reflection  than 
to  refraction.     The  planet's  spectrum  usually  shows  the  lines  of 
water  vapor,  but  they  may  be  telluric  (Art.  314). 

Lights  on  Dark  Portion.  —  Many  observers  have  also  reported 
faint  lights  as  visible  at  times  on  the  dark  portion  of  the  planet's 
disc.  These  cannot  be  accounted  for  by  reflection,  but  must  origi- 
nate on  the  planet's  surface  ;  they  recall  the  Aurora  Borealis  and 
other  electrical  manifestations  on  the  earth. 

574.  Satellites.  —  No   satellite   is   known,   although    in    the   last 
century  a  number  of   observers  at  various  times  thought   they  had 
found  one. 

In  most  cases  they  observed  small  stars  near  the  planet,  -which  we  can 
now  identify  by  computing  the  place  occupied  by  the  planet  at  the  date  of 
observation.  It  is  not,  however,  impossible  that  the  planet  may  have  some 
very  minute  and  near  attendants  like  those  of  Mars,  which  may  yet  be 
brought  to  light  by  means  of  the  great  telescopes  of  the  future,  or  by  pho- 
tography. Of  course  the  extreme  brilliance  of  the  planet,  and  the  fact  that 
the  necessary  observations  can  be  made  only  in  strong  twilight,  render  the 
discovery  of  such  objects,  if  they  exist,  very  difficult. 

575.  Transits.  —  Occasionally  Venus  passes   between   the   earth 
and  the  sun  at  inferior  conjunction,  giving  us  a  so-called  "transit." 

1  See  note  on  page  377. 


TRANSITS.  357 

She  is  then  visible  (even  to  the  naked  eye)  as  a  black  spot  on  the 
disc,  crossing  it  from  east  to  west. 

As  the  inclination  of  -the  planet's  orbit  is  nearly  3|°,  the  "  transit 
limit"  is  small  (about  4°),  and  the  transits  are  therefore  very  rare 
phenomena.  The  sun  passes  the  nodes  of  the  orbit  on  June  5  and 
December  7,  so  that  all  transits  must  occur  on  or  near  those  dates. 
When  Venus  crosses  the  sun's  disc  centrally,  the  duration  of  the 
transit  is  about  eight  hours.  Taking  the  mean  diameter  of  the  sun 
as  32',  or  g^-  of  a  circumference,  and  the  planet's  synodic  period 
as  584  days,  the  geocentric  duration  of  a  central  transit  should  be 
x  584^  Which  equals  0.332  days,  or  7h  58m. 


576.  Recurrence  of  Transits.  —  Five  synodic,  or  thirteen  sidereal, 
revolutions  of  Venus  are  very  nearly  equal  to  eight  years,  the  differ- 
ence being  only  a  little  more  than  one  day  ;  and  still  more  nearly,  in 
fact  almost  exactly,  243  years  are  equal  to  152  synodic,  or  395  side- 
real, revolutions.  If,  then,  we  have  a  transit  at  any  time,  we  may 
have  another  at  the  same  node  eight  years  earlier  or  later.  Sixteen 
years  before  or  after  it  would  be  impossible,  and  no  other  transit 
can  occur  at  the  same  node  until  after  the  lapse  of  two  hundred  and 
thirty-Jive  or  two  hundred  and  forty-three  years. 

If  the  planet  crosses  the  sun  nearly  centrally,  the  transit  will  not 
be  accompanied  by  another  at  an  eight-year  interval,  but  the  planet 
will  pass  either  north  or  south  of  the  sun's  disc,  at  the  conjunctions 
next  preceding  and  following.  If,  however,  as  is  now  the  case,  the 
transit  path  is  near  the  northern  or  southern  edge  of  the  sun,  then 
there  will  be  a  companion  transit  across  the  opposite  edge  of  the 
disc  eight  years  before  or  after.  Thus,  if  we  have  a  pair  of  June 
transits,  separated  by  an  eight-year  interval,  it  will  be  followed  by 
another  pair  at  the  same  node  in  243  years  ;  and  a  pair  of  December 
transits  will  come  in  about  halfway  between  the  two  pairs  of  June  tran- 
sits. After  a  thousand  years  or  so  from  the  present  time  the  transits 
will  cease  to  come  in  pairs,  as  they  have  been  doing  for  2000  years. 

577.  Transits  of  Venus  have  occurred  or  will  occur  on  the  following  dates  :  — 


Dec.  7,  1631         Dec.  9,  1874 
Dec.  4,  1639         Dec.  6,  1882 


June  5,  1761         June  8,  2004 
June  3,  1769         June  6,  2012 


1  If  Venus  (with  her  actual  rate  of  motion)  were  at  the  same  distance  from 

584d 
the  earth  as  from  the  sun,  the  duration  of  a  central  transit  would  be  7—;  but  at 

conjunction  she  is  nearer  in  the  ratio  of  277  to  723,  and  the  duration  is  corre- 
spondingly shortened. 

< 


358 


MAKS. 


The  special  interest  in  these  transits  consists  in  the  use  that  has  been 
made  of  them  for  the  purpose  of  finding  the  sun's 
parallax,  a  subject  which  will  be  discussed  later  on 
(Chap.  XVI.). 

The  first  observed  transit,  in  1639,  was  seen  by 
only  two  persons,  —  Horrox  and  Crabtree,  in  Eng- 
land. The  four  which  have  occurred  since  then 
have  been  extensively  observed  in  all  parts  of  the 
world  where  they  were  visible,  by  scientific  expeditions 
sent  out  for  the  purpose  by  different  nations.  The 
transits  of  1769  and  1882  were  visible  in  the  United  FIG.  ITS. 

States.     Fig.  173  shows  the  track  of  Venus  across   Transit  of  Venus  Tracks, 
the  sun's  disc  at  the  two  transits  of  1874  and  1882,, 

MARS. 

This  planet  is  also  prehistoric  as  to  its  discovery.  It  is  so  con- 
spicuous in  color  and  brightness,  and  in  the  extent  and  apparent 
capriciousness  of  its  movement  among  the  stars,  that  it  could  not 
have  escaped  the  notice  of  the  very  earliest  observers. 

578.  Orbit.  —  Its    mean    distance   from   the   sun   is   141,500000 
miles,  but  the  eccentricity  of  the  orbit  is  so  considerable   (0.093) 
that  the  distance  varies  about  13,000000  miles.     The  light  and  heat 
which  it  receives  from  the  sun  is  somewhat  less  than  half  of  that 
received  by  the  earth.     The  inclination  of  its  orbit  is  small,  1°  51'. 
The  planet's  sidereal  period  is  687  days,  or  ly  10|-mo,  which  gives 
it   an   average   orbital   velocity   of   fifteen    miles    per   second.      Its 
synodic  period  is  780  days,   or  2y  l|mo.      It  is  the  longest  in  the 
solar  system,  that  of  Venus  (584  days)  coming  next.     Of  the  780 
days,  it  moves  eastward   during  710,   and   retrogrades  during  70, 
through  an  arc  of  18°. 

579.  At  opposition  its  average  distance  from  the  earth  is  48,600- 
000  miles  (141,500000  miles  minus  92,900000   miles).     When  the 
opposition  occurs  near  the  planet's  perihelion,  this  distance  is  re- 
duced to  35,050000  miles ;  if  near  aphelion,  it  is  increased  to  over 
61,000000.     At  conjunction  the   average   distance   from   the    earth 
is  234,400000  miles  (141,500000  plus  92,900000). 

The  apparent  diameter  and  brilliancy  of  the  planet,  of  course,  vary 
enormously  with  these  great  changes  of  distance. 

If  we  put  R  for  the  planet's  distance  from  the  sun,  and  A  for  its  distance 
from  the  earth,  its  brightness,  neglecting  the  correction  for  phase,  should 


DIAMETER,    SURFACE,    AND   VOLUME.  359 

equal  — — .     We  find  from  this  that,  taking  the  brightness  at  conjunction 

as  unity  (at  which  time  the  planet  is  about  as  bright  as  the  pole-star),  it  is 
more  than  twenty-three  times  brighter  at  the  average  opposition,  and  fifty- 
three  times  brighter  if  the  opposition  occurs  at  the  planet's  perihelion.  At 
an  unfavorable  opposition  Mars,  as  has  been  said,  may  be  61,000000  miles 
distant,  and  its  brightness  then  is  only  about  twelve  times  as  great  as  at 
conjunction,  —  the  difference  between  favorable  and  unfavorable  oppositions 
being  more  than  four  to  one. 

These  favorable  oppositions  occur  always  in  the  latter  part  of  August  (at 
which  date  the  earth  passes  the  line  of  apsides  of  the  planet),  and  at  intervals 
of  fifteen  or  seventeen  years.  The  last  was  in  1907,  and  the  next  will  be  in 
1922.  A  reference  to  Fig.  159  will  show  how  great  is  the  difference  between 
the  planet's  opposition  distance  from  the  earth  under  varying  circumstances. 

580.  Diameter,  Surface,  and  Volume.  —  The  apparent  diameter  of 
the  planet  ranges  from  3". 6   at  conjunction,  to  25". 0  at  a  favorable 
opposition.     Its  real  diameter  is  very  closely  4200  miles,  — the  error 
may  be  twenty  miles  one  way  or  the  other.     This  makes  its  surface 
0.28,  and  its  volume  0.147  (equal  to  |)  of  the  earth's. 

581.  Mass,  Density,  and  Gravity. — Observations  upon  its  satel- 
lites   give  its  mass    as   JL    compared  with  that  of  the  earth.     This 
makes  its  density  0.73  and  superficial  gravity  0.38  ;  that  is,  a  body 
which  weighs  100  pounds  on  the  earth  would  have  a  weight  of  38 
pounds  on  the  surface  of  Mars. 

582.  Phases.  —  Since  the  orbit  of  the  planet  is  outside  that  of  the 

earth,  it  never  comes  between  us  and  the  sun, 
and  can  never  show  the  crescent  phase;  but  at 
quadrature  enough  of  the  unilluminated  portion 
is  turned  towards  the  earth  to  make  the  disc 
clearly  gibbous  like  the  moon  three  or  four  days 
from  full.  Fig.  174  shows  its  maximum  phase 
accurately  drawn  to  scale. 

FlQ  174  583.    The  "  Albedo  "  of  the  Planet.  —  Accord- 

Greatest  Phase  of  Mara      m£  to  Zollner's  observations  this  is  0.26,  which 
is  considerably^  higher  than  that  of  the  moon 
(£),  and  just  double  that  of  Mercury. 

584.  Rotation.  — The  planet's  time  of  rotation  is  24h  37m  228.67. 
This  very  exact  determination  has  been  made  by  Kaiser  and  Bak- 


360  MARS. 

huyzen,  by  comparing  drawings  of  the  planet  which  were  made  more 
than  200  years  ago  by  Hugyhens  with  others  made  recently. 

It  is  obvious  that  observations  made  a  few  days  or  weeks  apart  will 
give  the  time  or  rotation  with  only  approximate  accuracy.  Knowing 
it  thus  approximately,  we  can  then  determine,  without  fear  of  error, 
the  whole  number  of  rotations  between  two  observations  separated 
by  a  much  longer  interval  of  time.  This  will  give  a  second  and  closer 
approximation  to  the  true  period ;  and  with  this  we  can  carry  our 
reckoning  over  centuries,  and  thus  finally  determine  the  period  within 
a  very  minute  fraction  of  a  second.  The  number  given  is  not  uncer- 
tain by  more  than  -fa  of  a  second,  if  so  much. 

585.  The  Inclination  of  the  Planet's  Equator  to  the  Plane  of  its 
Orbit.  —  This   is  very  nearly  24°  according  to   Lowell,  not  very 
different  from  the  inclination  of  the  earth's  equator;  so  far,  therefore, 
as  depends  upon  that  circumstance,  its  seasons  should  be  substantially 
the  same  as  our  own. 

586.  Polar  Compression.  —  There  is  a  slight  but  sensible  flatten- 
ing of  the  planet  at  the  poles.     The  earlier  observers  found  for  the 
polar  compression  values  as  large  as  ?^,  and  even  y1^.     These  large 
values,  however,  are  inconsistent  with  the  existence  of  any  extensive 
surface  of  liquid  upon  the  planet,  and  more  recent  observations 
of  the  writer  show  the  polar  compression  to  be  about  ^fa.     This 
result  is  substantially  confirmed  by  the  still  later  measures  of  Lowell 
(who  gets  T^o)>  an(^  by  the  computations  of  H.  Struve  based  on  the 
perturbations  of  its  nearer  satellite.    It  is,  moreover,  almost  exactly 
what  would  be  expected  from  a  planet  constituted  as  we  suppose 
Mars  to  be. 

587.  Telescopic  Appearance  and  Surface-Markings.  —  The  fact 

that  we  are  able  to 
determine  the  time  of 
rotation  so  accurately 
of  course  implies  the 
existence  of  identifiable 
markings  upon  the  sur- 
face. Viewed  through 
a  powerful  telescope, 

FIG.  175.-Telescopic  Views  of  Mars.  the  planet>s   disc>   as   a 

whole,  is  ruddy,  or  orange-colored,  and  is  specially  bright  around 
the  limb,  but  not  at  the  "  terminator,"  if  there  is  any  considerable 


TELESCOPIC    APPEARANCE.  361 

phase.  The  central  portions  of  the  disc  present  greenish  and 
purplish  patches  of  shade,  for  the  most  part  not  sharply  defined, 
though  some  of  the  markings  have  outlines  reasonably  distinct.  On 
watching  the  planet  for  only  a  few  hours  even,  the  markings  pass 
on  across  the  disc,  and  are  replaced  by  others.  Some  of  them  are 
permanent,  and  recur  at  regular  intervals  with  the  same  form  and 
appearance,  while  others  appear  to  be  only  clouds  which  for  a  time 
veil  the  surface  below,  and  then  clear  away.  But  these  are  extremely 
rare  as  compared  with  clouds  upon  the  earth. 

The  most  noticeable  features  may  be  divided  broadly  into  three 
classes,  — 

First,  white  patches,  two  of  which,  near  the  planet's  poles,  are 
usually  conspicuous,  and  are  generally  supposed  to  be  sheets  of 
snow  and  ice,  since  they  behave  just  as  would  be  expected  if  such 
were  the  case.  During  the  planet's  northern  summer  the  northern 
cap  dwindles  away,  while  the  southern  one  rapidly  increases,  and 
vice  versa  during  the  southern  summer.  At  times  the  southern  cap 
is  more  than  1800  miles  across,  and  four  or  five  months  later  it 
sometimes  entirely  disappears. 

Second,  patches  of  bluish  gray  or  greenish  shade,  covering  usually 
about  three-eighths  of  the  planet's  surface.  They  lie  for  the  most 
part  in  the  southern  hemisphere  and  mainly  in  the  equatorial  region, 
forming,  in  a  small  telescope,  a  sort  of  darkish  belt  around  the 
planet.  These  until  very  lately  have  been  almost  universally  ad- 
mitted to  be  sheets  of  water,  and  have  received  the  names  of  "seas/7 
"gulfs,"  etc.  But  recent  observations  make  this  doubtful,  and 
suggest  that  they  are  more  probably  regions  covered  with  vegeta- 
tion, and  that  no  great  bodies  of  water  exist  on  Mars. 

Third,  extensive  regions  of  various  shades  of  orange,  covering  more 
than  half  the  surface,  especially  in  the  northern  hemisphere ;  it  is 
generally  agreed  that  these  are  land,  probably  deserts  of  sand  and 
rock. 

Fig.  175  gives  an  idea  of  the  planet's  general  telescopic  appear- 
ance, though  with  no  attempt  at  minute  accuracy.  It  fails  also  in 
not  showing  how  all  the  markings  fade  out  at  some  distance  from 
the  brilliant  edge  of  the  disc,  an  effect  doubtless  due  to  the  planet's 
atmosphere. 

588.  Recent  Discoveries.  The  Canals  and  their  "Gemination." 
Oases.  Seasonal  Changes.  —  a.  Besides  the  conspicuous  markings 
already  mentioned,  there  are  others,  difficult  to  observe,  but  of  great 
interest  and  significance.  In  1877  and  1879  Schiaparelli  announced 


362  MAES. 

the  discovery  of  a  great  number  of  fine,  dark  straight  lines,  or 
"canals,"  as  he  called  them,  crossing  the  ruddy  portions  of  the 
disc  in  all  directions ;  and  in  1881  he  announced  further  that  many 
of  these  became  double  at  times,  like  the  two  parallel  tracks  of  a 
railway.  His  observations  have  since  been  confirmed  and  added  to 
by  various  eminent  astronomers  in  Europe  and  America,  especially 
by  Perrotin  at  Nice  and  Lowell  in  Arizona.  But  others,  equally 
eminent  and  apparently  under  equally  favorable  conditions,  fail  to 
see  the  reported  features.  At  present,  from  numerous  experiments 
recently  made  upon  maps  and  models  viewed  at  different  distances, 
it  seems  probable  that  illusions  enter  into  the  case  to  some  extent, — 
illusions  resulting  from  the  unconscious  misinterpretation  of  actual 
markings  imperfectly  seen.  The  so-called  "  canals  "  doubtless  exist, 
though  probably  not  exactly  as  represented  upon  the  map.  There 
are,  however,  special  reasons  to  suspect  the  reality  of  their  "  gemi- 
nation." And  yet  it  is  by  no  means  impossible  that  time  may  bring 
a  complete  confirmation  of  Schiaparelli's  results. 

b.  According  to  several  observers,  especially  the  Flagstaff  astron- 
omers, the  canals  are  not  limited  to  the  ruddy  portions  of  the  sur- 
face, but  in  some  cases  extend  across  the  dusky  regions  also.     This 
observation,  if  correct,  is  of  great  importance  in  showing  that  the 
so-called  "  seas "  cannot  be  bodies  of  water.     But  it  is  difficult,  and 
needs  to  be  more  fully  confirmed  before  final  acceptance. 

c.  At  the  intersection  of  the  canals  (of  which  over  180  are  now 
catalogued  and  located  on  Mr.  Lowell's  map  of  the  planet)  small, 
round  dark  spots  are  observed  at  certain  times.    These,  at  first  called 
"lakes,"  are  now  regarded  by  him  as  "oases" 

d.  It  is  found  also  that  the  extent  and  darkness  of  the  so-called 
"seas  "  varies  greatly  with  the  Martial  seasons.    Speaking  generally, 
the  "seas"  of  each  hemisphere  are  darker  and  larger  at  the  time 
when  their  respective  polar  caps  begin  to  shrink,  and  they  become 
smaller,  and  more  definite  in  outline  when  the  caps  are  on  the 
point  of  vanishing. 

e.  Mr.  Lowell's  observations  appear  to  show  that  there  are  few 
if  any  high  mountain  peaks  or  ranges,  though  there  are  indications 
of  a  few  elevations  perhaps  two  or  three  thousand  feet  in  height. 
The  planet's  surface  seems  to  be  remarkably  level  as  compared  with 
the  earth's. 

589.  Atmosphere  and  Temperature.  —  At  one  time  it  was  sup- 
posed that  Mars  possessed  a  very  dense  atmosphere  which  gave  it 
its  ruddy  color ;  but  considerations  based  on  the  planet's  low 
surface  gravity  (not  quite  38  per  cent  of  the  earth's)  as  well  as  the 


ATMOSPHERE   AND   TEMPERATURE.  363 

direct  evidence  of  observation,  show  that  this  cannot  be  the  case. 
There  is  no  doubt,  however,  that  it  has  some  atmosphere,  since  the 
occasional  presence  of  thin  veils  of  cloud,  as  well  as  the  deposition 
and  dissipation  of  the  polar  caps,  can  be  explained  only  by  its 
presence;  but  its  height  and  density  must  be  much  less  than  that 
of  our  own.  Some  of  the  earlier  spectroscopic  observers  considered 
that  they  had  found  certain  evidence  of  the  presence  of  water-vapor, 
but  later  observations,  especially  those  of  Campbell  at  the  Lick 
Observatory,  render  the  case  doubtful,  to  say  the  least. 

As  to  the  temperature  of  Mars,  we  have  no  certain  knowledge. 
On  the  one  hand  we  know  that  on  account  of  the  planet's  distance 
from  the  sun  the  intensity  of  solar  radiation  upon  its  surface  must 
be  less  than  here  in  the  ratio  of  1  to  (1.524)2,  i.e.,  only  about  43 
per  cent  as  great  as  with  us:  its  "solar  constant"  must  be  less 
than  10  calories  against  our  21.  Then,  too,  the  low  density  of  its 
atmosphere,  probably  less  at  the  planet's  surface  than  on  the  tops 
of  our  highest  mountains,  would  naturally  assist  to  keep  down  the 
temperature  to  a  point  far  below  the  freezing-point  of  water.  But 
on  the  other  hand  things  certainly  look  as  if  the  polar  caps  were  really 
masses  of  snow  and  ice  deposited  from  vapor  in  the  planet's  atmos- 
phere, and  as  if  these  actually  melted  during  the  Martian  summer, 
sending  floods  of  water  through  the  channels  provided  for  them,  and 
causing  the  growth  of  vegetation  along  their  banks.  We  are  driven, 
therefore,  to  suppose  either  that  the  planet  has  sources  of  heat, 
internal  or  external,  which  are  not  yet  explained ;  or  else,  as  long 
ago  suggested,  that  the  polar  "snow"  may  possibly  be  composed 
of  something  else  than  frozen  water.  The  problem  is  a  perplexing 
one,  and  it  is  earnestly  to  be  hoped  that  before  very  long  we  may 
come  into  possession  of  some  heat  measurer  sufficiently  delicate  to 
give  us  direct  evidence  as  to  the  warmth  or  coldness  of  the  planet's 
surface.  (See  also  note  to  Art.  589**.) 

589*.  Speculations  of  Flammarion  and  Lowell. — These  astronomers, 
and  many  others,  practically  ignore  the  temperature  difficulty,  and  unhesitat- 
ingly assume  that  the  polar  caps  are  composed  of  snow  and  ice,  that  they 
melt  in  spring  and  summer,  and  that  the  water  thus  liberated  makes  its  way 
towards  the  equator  over  the  planet's  mountainless  plains,  partially  obscur- 
ing for  several  weeks*  the  well-known  features  which  at  other  times  are  con- 
spicuous. According  to  Mr.  Lowell  the  dark  regions  formerly  supposed  to 
be  seas  are  regions,  possibly  marshy,  more  or  less  covered  with  vegetation, 
while  the  ruddy  portions  are  Saharan  deserts  intersected  with  water-channels, 
which  he  regards  as  artificial  (in  part,  at  least)  and  arranged  for  purposes  of 
irrigation.  When  the  water  reaches  these,  verdure  springs  up  along  their 


364 


MARS. 


course,  and  these  streaks  of  vegetation  are  what  we  recognize  as  the  "canals," 
the  water-channels  themselves  being  far  too  narrow  to  be  visible  in  our  tele- 
scopes. When  they  cross  each  other  "  oases "  are  formed.  As  to  the 
"  gemination  "  of  the  canals,  no  clear  explanation  appears  as  yet,  though 
suggestions  have  been  offered  that  it  may  be  due  to  some  mode  of  growth, 
or  some  treatment  of  the  crops  produced  by  the  irrigation. 

589**.  Habitability  of  Mars.  —  It  may  be  said  with  some  confidence 
that  on  Mars  the  conditions,  different  as  they  must  be  from  our  own,  are 
still  more  nearly  earthlike  than  on  any  other  of  the  heavenly  bodies  which 
we  can  see  with  our  present  telescopes. 

And  yet,  as  already  pointed  out,  unless  the  planet  has  unknown  sources 
of  heat,  the  temperature1  must  be  too  low  to  permit  anything  like  terrestrial 
life.  Nor,  with  one  possible  exception,  is  there  the  slightest  evidence  of  the 
existence  of  intelligent  beings  upon  it.  Mr.  Lowell  argues  from  the  straight- 
ness  of  the  canals  (some  of  them  over  2000  miles  in  length),  and  from  the 
accuracy  with  which  several  of  them  converge  at  certain  oases,  that  they 


FIG.  176.  —  Mars,  Lowell, 


must  have  been  intelligently  engineered.  (See  Fig.  176.)  If  his  observations 
on  this  point  are  not  illusions  based  on  misconception  his  conclusion  is  per- 
haps not  unnatural  though  by  no  means  necessary ;  but  where  seeing  is 
difficult  at  best,  it  is  easy  for  one  to  imagine  that  he  sees  what  he  thinks  he 
ought  to  see. 

589***.  Maps  of  Mars.  —  Numerous  maps  of  the  planet  have 
been  constructed  by  various  astronomers  since  the  first  was  drawn 
by  Maedler  in  1830.  Fig.  176*  is  from  one  published  by  Schiaparelli 
in  1888,  and  shows  most  of  his  canals,  and  the  gemination  of  such 
of  them  as  exhibit  that  phenomenon.  There  can  be  no  doubt  as  to 

1  Mr.  Lowell  concludes  from  an  elaborate  mathematical  investigation  that  the 
mean  temperature  of  the  planet  is  about  48°  F.  or  9°  C.  This  is  much  higher 
than  that  considered  in  Art.  589.  The  mean  temperature  of  the  earth  is  usually 
taken  as  60°  F. 


CHART    OF   MARS. 


365 


366  MAES. 

the  substantial  correctness  of  the  principal  features,  but  as  to  minor 
details  there  may  be  considerable  uncertainty ;  nor  must  it  be  for- 
gotten that  certain  important  features  notably  change  their  size  and 
appearance  with  the  progress  of  the  planet's  seasons.  In  1904-5 
Lampland  at  Professor  Lowell's  Flagstaff  Observatory  obtained  pho- 
tographs of  the  planet,  which,  for  the  first  time,  show  nearly  all  the 
features  visible  in  ordinary  telescopes.  In  1907  Professor  Todd, 
while  in  charge  of  an  expedition  sent  to  northern  Chile  by  Professor 
Lowell,  obtained  photographs  which  show  many  of  the  canals  and 
suggestions  of  "  gemination." 

590.  Satellites.  —  There  are  two  satellites  which  were  discovered 
in  August,  1877,  by  Professor  Hall  at  Washington,  with  the  then 
new  26-inch  telescope.     They  are  exceedingly  minute,  and  can  be 
seen  only  with  the  most  powerful  instruments.     The  outer  one, 
Deimos,  is  at  a  distance  of  14600  miles  from  the  centre  of  the 
planet,  and  has  a  period  of  30h  18m,  while  the  inner  one,  Phobos, 
is  at  a  distance  of  only  5800  miles,  and  its  month  is  but  7h  39m  long, 
not  one-third  of  the  day  of  Mars.     Owing  to  this  fact  it  rises  in  the 
west  every  night  for  the  "  Marticoli  "  (if  there  are  any  people  there) 
and  sets  in  the  east,  after  about  5-J-h. 

Deimos  does  not  do  this  ;  it  rises  in  the  east  like  other  stars,  but 
its  orbital  eastward  motion  among  the  stars  is  so  nearly  equal  to  its 
diurnal  motion  westward,  that  it  is  nearly  132  hours  between  two 
successive  risings.  This  is  more  than  four  of  its  months,  so  that  it 
undergoes  all  its  changes  of  phase  four  times  in  the  interval. 

Of  course,  both  the  satellites  are  frequently  eclipsed.  Their 
orbits  appear  to  be  exactly  circular,  and  they  move  exactly  in  the 
plane  of  the  planet's  equator  ;  and  they  keep  so,  maintained  in  their 
relation  to  the  equator  by  the  action  of  the  "  equatorial  bulge  "  tfpon 
the  planet. 

591.  As  givers  of  moonlight  they  do  not  amount  to  much.    Their 
diameters  are  too  small  to  be  measured  with  any  micrometer ;  but 
from  their  apparent  "  magnitude  "  (i.e.,  brightness),  as  seen  from  the 
earth,  and  assuming  that  their  surfaces  have  the  same  reflective 
power  as  that  of  the  planet,  Professor  Pickering  has  estimated  the 
diameter  of  Phobos,  which  is  the  larger  one,  as  about  seven  miles, 
and  that  of  Deimos,  as  five  or  six.     The  light  given  by  Phobos  to 
the  inhabitants  of  Mars  would  be  about  -fa  of  our  moonlight;  that 
of  Deimos  about  T^Vir-     Mr.  Lowell's  estimates  of  their  diameters 
are  considerably  greater,  — 10  miles  for  Deimos,  and  25  for  Phobos. 


THE   ASTEROIDS,    OK   MINOR    PLANETS.  367 

The  period  of  Phobos  is  by  far  the  shortest  period  in  the  solar 
system.  Its  rapidity  of  revolution  raises  important  questions  as  to 
the  theory  of  the  development  of  the  solar  system,  and  requires 
modification  of  the  views  which  had  been  held  up  to  the  time  of 
their  discovery.  If  the  nebular  hypothesis  is  true,  a  shortening  of 


FIG.  177.  —  Orbits  of  the  Satellites  of  Mars. 

the  satellite's  period,  or  a  lengthening  of  the  planet's  day,  must  have 
occurred  since  the  satellite  came  into  being,  since  that  hypothesis 
will  not  account  for  the  existence  of  a  satellite  having  a  period 
shorter  than  the  diurnal  rotation  of  its  primary.  Fig.  177  is  a 
diagram  of  the  satellite  orbits  as  they  appeared  from  the  earth  in 
1888.  It  is  reduced  from  the  American  Nautical  Almanac  for  that 
year. 

THE  ASTEROIDS,  OR  MINOR  PLANETS. 

592.  These  are  a  group  of  small  planets  circulating  in  the  space 
between  Mars  and  Jupiter.  The  name  "asteroid"  was  suggested 
by  Sir  William  Herschel  early  in  the  century,  when  the  first  ones 
were  discovered.  The  later  term  planetoid,  is  preferred  by  some. 

It  was  very  early  noticed  that  there  is  a  break  in  the  series  of  the 
distances  of  the  planets  from  the  sun.  Kepler,  indeed,  at  one  time 
thought  he  had  discovered  the  true  law  \  and  the  real  reason  why 

1  His  supposed  law  was  as  follows  :  Imagine  the  sun  surrounded  by  a  hollow 
spherical  shell,  on  which  lies  the  orbit  of  the  earth.  Inside  of  this  shell  inscribe 
a  regular  icosahedron  (the  twenty-sided  regular  solid),  and  within  that  inscribe  a 
second  sphere.  This  sphere  will  carry  upon  it  the  orbit  of  Venus.  Inside  of  the 
sphere  of  Venus  inscribe  an  octahedron  (the  eight-sided  solid),  and  the  sphere 
which  fits  within  it  will  carry  Mercury's  orbit.  Next,  working  outwards  from  the 


368  THE   MINOR    PLANETS. 

the  planets7  distances  are  what  they  are.  This  theory  of  his 
was  broached  twenty-two  years,  however,  before  he  discovered  the 
harmonic  law,  and  he  probably  abandoned  it  when  he  discovered 
the  elliptical  form  of  the  planets'  orbits.  At  any  rate,  in  later  life 
he  suggested  that  it  was  likely  that  there  was  a  planet  between 
Mars  and  Jupiter  too  small  to  be  seen. 

The  impression  that  such  a  planet  existed  gained  ground  when 
Bode,  in  1772,  published  the  law  which  bears  his  name,  and  it  was 
still  further  deepened  when,  nine  years  later,  in  1781,  Uranus  was 
discovered,  and  the  distance  of  the  new  planet  was  found  to  con- 
form to  Bode's  law.  An  association  of  twenty-four  astronomers, 
mainly  German,  was  immediately  formed  to  look  for  the  missing 
planet,  who  divided  the  zodiac  between  them  and  began  the  work. 
Singularly  enough,  however,  the  first  discovery  was  made,  not  by  a 
member  of  this  association,  but  by  Piazzi,  the  Sicilian  astronomer  of 
Palermo,  who  was  then  engaged  upon  an  extensive  star  catalogue. 
On  January  1,  1801,  he  observed  a  seventh-magnitude  star  which 
by  the  next  evening  had  unquestionably  moved,  and  kept  on  moving. 
He  observed  it  carefully  for  some  six  weeks,  when  he  was  taken  ill ; 
before  he  recovered,  it  had  passed  on  towards  superior  conjunction, 
and  was  lost  in  the  rays  of  the  sun.  He  named  it  Ceres,  after  the 
tutelary  goddess  of  Sicily. 

When  at  length  the  news  reached  Germany  in  the  latter  part  of  March 
it  created  a  great  excitement,  and  the  problem  now  was  to  rediscover  the 
lost  planet.  The  association  of  planet-hunters  began  the  search  in  Septem- 
ber, as  soon  as  its  elongation  from  the  sun  was  great  enough  to  give  any 
prospect  of  success.  During  the  summer  Gauss  devised  his  new  method 
of  computing  a  planetary  orbit,  and  computed  the  ephemeris  of  its  path. 
Very  soon  after  receiving  his  results,  Baron  Von  Zach  rediscovered  Ceres 
on  December  31,  and  Dr.  Olbers  on  the  next  day,  just  one  year  after  it  was 
first  found  by  Piazzi. 

earth's  orbit,  circumscribe  around  the  earth's  sphere  a  dodecahedron,  circum- 
scribing around  it  another  sphere,  and  this  will  carry  upon  it  the  orbit  of  Mars. 
Around  the  sphere  of  Mars  circumscribe  the  tetrahedron,  or  the  regular  pyra- 
mid. The  corners  of  this  solid  project  very  far,  so  that  the  sphere  circumscribed 
around  the  tetrahedron  will  be  at  a  very  great  distance  from  the  sphere  of  Mars. 
It  carries  the  orbit  of  Jupiter.  Finally,  the  cube,  or  hexahedron,  circumscribed 
around  the  orbit  of  Jupiter,  gives  us  in  the  same  way  the  orbit  of  Saturn.  We 
thus  obtain  a  series  of  distances  not  enormously  incorrect  (though  by  no  means 
agreeing  with  fact  even  as  closely  as  does  Bode's  law) ;  and,  moreover,  the  theory 
had  the  great  advantage  to  Kepler's  mind  of  accounting  for  the  fact  that  there 
are  (so  far  as  was  then  known)  but  seven  planets,  there  being  possible  but  five 
regular  solids. 


METHOD   OF   SEARCH.  369 

593.  In  March,  1802,  Dr.  Gibers,  who  in  looking  for  Ceres  had 
carefully  examined  the  small  stars  in  the  constellation  of  Virgo,  on 
going  over  the  ground  again,  found  a  second  planet,  which  he  named 
Pallas,  a  body  of  about  the  same  brightness  as  Ceres.     Two  having 
now  been  found,  and  Pallas  having  a  very  eccentric  and  much  inclined 
orbit,  he  conceived  the  idea  that  they  were  fragments  of  a  broken 
planet,  and  that  other  planets  of  the  same  group  could  probably  be 
found  by  searching  near  the  intersection  of  their  two  orbits.     Juno, 
the  third,  was  discovered  by  Harding  at  Lilienthal  (Schroter's  obser- 
vatory) in  1804;  and  Vesta,  the  largest  and  brightest  of  the  whole 
group  (sometimes  visible  to  the  naked  eye),  was  found  by  Olbers 
himself  in  1807.     The  search  was  kept  up  for  several  years  after 
this,  but  no  more  planets  were  found  because  they  did  not  look  after 
small  enough  stars. 

The  fifth,  Astrsea,  was  discovered  in  1845  by  Hencke,  an  amateur 
astronomer  who  for  fifteen  years  had  been  engaged  in  studying  the 
smaller  stars  in  hopes  of  just  the  reward  he  captured.     In  1846  no 
asteroid  was  found  (the  discovery  of  Neptune  was  glory  enough  for 
that  year),  but  in  1847  three  more  were  brought  to  light ;  and  since 
then  not  a  year  has  passed  without  adding  from  one  to  a  hundred 
to  the  number.     In  September,  1907,  the  list  given  by  the  "  Kechen- 
institut"  counted  up  635,  duly  identified  and  "numbered/7  with 
about  fifty  more  not  then  sufficiently  observed  to  warrant   com- 
plete recognition.     Fresh  discoveries  are  made  continually,  though 
the  new  asteroids  are  mostly  very  small,  —  stars  fainter  than  the 
twelfth  magnitude,  which  require  a  large  telescope  to  make  them  even 
visible.    All  the  brighter  ones  have  evidently  been  already  picked  up. 

594.  Method  of  Search.  —  Formerly  the  search  for  these  objects 
was  conducted  by  making  special  star  charts  of  certain  regions  near 
the  ecliptic  selected  by  the  "asteroid-hunter,"  and  afterwards  com- 
paring the  chart  with  the  heavens,  when  interlopers  would  at  once 
be  detected,  and  their  nature  determined  by  their  motion.     The 
operation  was  very  laborious. 

During  1891  a  new  method  of  search  was  introduced.  By  photo- 
graphing1 a  portion  of  the  heavens  with  a  camera  of  wide  field, 
mounted  equatorially  and  moved  by  clock-work,  pictures  are  ob- 
tained in  which  any  planets  present  can  be  easily  distinguished  by 
their  motion  during  the  two  or  three  hours  during  which  the  exposure 
of  the  plate  is  continued  ;  while  the  images  of  stars  are  round,  if  the 
clock-work  runs  correctly,  the  planets  are  apparently  elongated  into 
streaks. 

i  See  Addendum  D,  page  580  i. 


870  THE   MINOR   PLANETS. 

Of  course,  great  care  must  be  taken  to  be  sure  that  the  object  discovered 
is  a  new  planet,  and  not  one  of  the  multitude  already  known.  Generally  it 
is  possible  to  decide  very  quickly  which  of  the  known  planets  will  be  in 
the  neighborhood,  and  a  rough  computation  will  commonly  decide  at  once 
whether  the  planet  is  new.  Not  always,  however,  and  mistakes  in  this 
regard  are  not  very  unusual. 

The  known  asteroids  have  been  discovered  by  comparatively  a  few  ob- 
servers. Four  persons,  working  by  the  old  method,  have  discovered  more 
than  twenty  each.  Palisa  of  Vienna  has  discovered  83  ;  the  late  Dr.  Peters  of 
Clinton,  N.Y.,  52  ;  Luther  of  Dusseldorf,  24 ;  and  the  late  Professor  Watson 
of  Ann  Arbor,  22.  More  recently  Charlois  of  Nice  has  discovered  over  100, 
nearly  all  by  the  photographic  method  ;  and  Max  Wolf  of  Heidelberg,  and 
his  assistants  a  still  larger  number,  all  by  photography. 

These  minor  planets  are  mostly  named,  the  names  being  derived 
from  mythology  and  legend.  They  are  also  designated  by  numbers, 
and  the  symbol  for  each  planet  is  the  number  written  in  a  circle.  Thus, 
for  Ceres  the  symbol  is  (i) ;  for  Hilda,  (153) ;  and  so  on. 

A  full  list  of  them,  with  the  elements  of  their  orbits,  is  published  bien- 
nially in  the  "  Annuaire  du  Bureau  des  Longitudes,"  Paris. 

595.  Their  Orbits.  —  The  mean  distance  of  the  different  asteroids 
from  the  sun  varies  greatly  and  the  periods  are  correspondingly 
different.     Hungaria,  (434),  has  the  smallest  mean  distance  (except- 
ing the  recently  discovered  Eros  *),  viz.,  1.94  or  180  000000  miles, 
with  a  period  of  2y  9mo.     Thule,  (279),  is  the  remotest,2  with  a  mean 
distance  of  4.30  or  400000000  miles,  and  a  period  of  8*  313d.     Ac- 
cording to  Svedstrup,  the  mean  distance  of  the  "  mean  asteroid " 
is  2.65  (246000000  miles),  and  its  period  about  4£  years.     Its  dis- 
tance from  the  earth  at  time  of  opposition  would  be,  of  course,  1.65, 
or  153  000000  miles. 

The  inclinations  of  their  orbits  average  about  8°;  but  Pallas,  ®, 
has  an  inclination  of  35°,  and  Euphrosyne,  (31),  of  26-j-0. 

Several  of  the  orbits  are  extremely  eccentric.  ^Ethra,  (132),  has 
an  almost  cometary  eccentricity  of  0.38,  and  over  a  dozen  others 
have  eccentricities  exceeding  0.30.  They  are  distributed  quite 
unequally  in  the  range  of  distance,  there  being,  as  Kirkwood  has 
pointed  out,  very  few  at  such  distances  that  their  periods  would  be 
exactly  commensurable  with  that  of  Jupiter. 

596.  Diameter  and  Surface.  —  These  bodies  are  so  small  that 
micrometrical  measurements  upon  them  are  extremely  difficult,  and 
until  very  recently  our  estimates  of  their  probable  size  have  been 
based  merely  upon  their  brightness.     Pickering,   by  photometric 

1  See  page  377  for  note  on  Eros.          2  See  Addendum  D,  page  580  i. 


AGGREGATE   MASS.  371 

methods,  and  assuming  an  "  albedo  "  the  same  as  that  of  Mars,  found 
for  Vesta  (the  only  one  ever  visible  to  the  naked  eye)  a  diameter  of 
319  miles.  In  1894-95,  however,  Mr.  Barnard  with  the  great  Lick 
telescope  succeeded  in  making  micrometric  measures  of  the  discs  of 
the  four  brightest  with  the  following  rather  surprising  results :  — 
Diameter  of  Ceres,  485  miles  ;  of  Pallas,  304  ;  of  Juno,  118  ;  and  of 
Vesta,  243.  But  the  percentage  of  probable  error  must  be  pretty  large. 
The  surprise  lies,  of  course,  in  the  great  contrast  of  albedo  between 
Vesta  and  the  other  three.  As  to  the  rest  of  the  family,  it  is  hardly 
possible  that  any  one  of  them  can  be  as  much  as  100  miles  in 
diameter,  and  the  smallest  are  probably  less  than  ten  miles  through, 
—  nothing  more  than  "mountains  broke  loose.'7 

597.  Mass,  Density,  etc.  —  As  to  the  individual  masses  and  densi- 
ties we  have  no  certain  knowledge.     It  is  probable  that  the  density 
does  not  differ  much  from  the  density  of  the  crust  of  the  earth,  or 
the  mean  density  of  Mars.     If  this  is  so,  the  mass  of  Ceres  might 
possibly  be  as  great  as  ^V<y  of  the  earth.     On  such  a  planet  the 
force  of  superficial  gravity  would  be  about  ^Vd  of  gravity  on  the 
earth,  and  a  body  projected  from  the  surface  with  a  velocity  of 
about  2500  feet  a  second  —  that  of  an  ordinary  rifle-ball  —  would 
fly  off  into  space  and  never  return  to  the  planet,  but  would  circulate 
around  the  sun  as  a  planet  on  its  own  account.     On  the  smallest 
asteroids,  with  a  diameter  of  about  ten  miles,  it  would  be  quite 
possible  to  throw  a  stone  from  the  hand  with  velocity  enough  to 
send  it  off  into  space. 

598,  Aggregate  Mass.  —  Although  we  can  only  estimate  very 
roughly  the  masses  of  the  individual  members  of  the  flock,  it  is  pos- 
sible to  get  some  more  certain  knowledge  of  their  aggregate  mass. 
Leverrier  from  the  motion  of  the  line  of  apsides  of  the  orbit  of  Mars 
demonstrated  that  the  whole  amount  of  matter  thus  distributed  in 
the  space  between  Mars  and  Jupiter  cannot  exceed  about  one-fourth 
of  the  mass  of  the  earth.     A  still  later  computation  by  Kavene  in 
1896,  indicates  a  total  mass  only  about  T|y  as  great  as  the  earth's. 

The  united  masses  of  those  which  are  already  known  would  make  only  a 
very  small  fraction  of  such  a  body.  Up  to  August,  1880,  the  united  bulk  of 
the  asteroids  then  discovered  was  estimated  at  -rfcy  part  of  the  earth's  bulk, 
with  a  mass  probably  about  ^TO  of  the  earth's.1  Presumably,  therefore, 
the  number  of  these  bodies  remaining  undiscovered  is  exceedingly  great  — 

1  Barnard's  measures  (Art.  596)  would  increase  this  estimate  of  bulk  and  mass, 
but  would  not  seriously  affect  the  general  conclusion. 


372  THE   MINOR   PLANETS. 

to  be  counted  by  thousands,  if  not  by  millions.  Most  of  them,  of  course, 
must  be  much  smaller  than  those  which  are  already  known. 

599.  Forms,  Variations  of   Brightness,  and  Atmosphere.  —  We 

have  as  yet  only  scanty  knowledge  on  this  point,  but  Dr.  Olbers 
observed  in  Vesta  certain  fluctuations  in  her  brightness  which 
seemed  to  him  to  indicate  that  she  is  not  a  globe,  but  an  angular 
mass,  —  a  splinter  of  rock.  This,  however,  is  not  confirmed  by  the 
more  recent  photometric  observations  of  Miiller  or  Pickering. 

Since  1900,  however,  it  has  been  discovered  by  photometric  methods  that 
some  of  these  bodies  show  regular  variations  of  brightness  recurring  at 
intervals  of  from  5  to  8  hours,  due  probably  to  axial  rotation.  Iris,  (7) ; 
Sirona,  (Tie) ;  Hertha,  (135) ;  Tercidina,  (345) ;  Eros,  (433) ;  several  others  are 
already  known  to  behave  in  this  way,  and  yet  others  are  suspected. 

600.  Origin.  —  With  respect  to  this  we  can  only  speculate.     Two 
views  have  been  held,  as  has  been  already  intimated.    One  is,  that  the 
material,  which  according  to  the  nebular  hypothesis  ought  to  have 
been  concentrated  to  form  a  single  planet  of  the  class  to  which  the 
earth  belongs,  has  failed  to  be  so  collected,  and  has  formed  a  flock 
of  small  separate  masses.     It  has  been  generally  believed  that  the 
matter  which  at  present  forms  the  planets  was  once  distributed  in 
rings,  like  the  rings  of  Saturn.     If  so,  this  ring,  or  meteoric  swarm, 
would  necessarily  suffer  violent  perturbations  from  the  nearness  of 
the  enormous  planet  Jupiter,  and  so  would  be  under  very  different 
conditions  from  any  of  the  other  rings.     This,  as  Peirce  has  shown, 
might  account  for  its  breaking  up  into  many  fragments. 

The  other  view  is  that  a  planet  about  the  size  of  Mars  has  broken 
to  pieces.  It  is  true,  as  has  been  often  urged,  that  this  theory  in  its 
original  form,  as  presented  by  Olbers,  cannot  be  correct.  No  single 
explosion  of  a  planet  could  give  rise  to  the  present  assemblage  of 
orbits,  nor  is  it  possible  that  even  the  perturbations  of  Jupiter  could 
have  converted  a  set  of  orbits  originally  all  crossing  at  one  point 
(the  point  of  explosion)  into  the  present  tangle.  The  smaller  orbits 
are  so  small  that  however  turned  about  they  lie  wholly  inside  the 
larger,  and  cannot  be  made  to  intersect  them.  If,  however,  we  admit 
a  series  of  explosions,  this  difficulty  is  removed ;  and  if  we  grant 
an  explosion  at  all,  there  seems  to  be  nothing  improbable  in  the 
hypothesis  that  the  fragments  formed  by  the  bursting  of  the  parent 
mass  would  carry  away  within  themselves  the  same  forces  and 
reactions  which  caused  the  original  bursting ;  so  that  they  themselves 
would  be  likely  enough  to  explode  at  some  time  in  their  later  history. 

At  present  opinion  is  divided  between  these  two  theories. 


INTRA-MEKCURIAL   PLANETS.  373 

601.  The  number  of  these  bodies  already  known  is  so  great,  and  the 
prospect  for  the  future  is  so  indefinite,  that  astronomers  are  at  their  wits' 
end  how  to  take  care  of  this  numerous  family.     To  compute  the  orbit  and 
ephemeris  of  one  of  these  little  rocks  is  more  laborious  (on  account  of  the 
great  perturbations  produced  by  Jupiter)  than  to  do  the  same  for  one  of  the 
major  planets  ;    and  to  keep  track  of  such  a  minute  body  by  observation  is 
far  more  difficult.     Until  recently,  the  German  Jahrbuch  has  been  publish- 
ing the  ephemerides  of  such  as  came  within  the  range  of  observation  each 
year ;  but  this  cannot  be  kept  up  much  longer,  and  the  probability  is  that 
hereafter  only  the  larger  ones,  or  those  which  present  some  remarkable 
peculiarity  in  their  orbits,  will  be  followed  up.     One  little  family  of  them, 
however,  is  "endowed."     Professor  Watson,  at  his  death,  left  a  fund  to  the 
American  National  Academy  of  Sciences  to  bear  the  expense  of  taking  care 
of  the  twenty-two  which  he  discovered. 

INTRA-MERCURIAL  PLANETS  AND   THE   ZODIACAL  LIGHT. 

It  is  not  at  all  improbable  that  there  are  masses  of  matter  revolv- 
ing around  the  sun  within  the  orbit  of  Mercury. 

602.  Motion  of  the  Perihelion  of  Mercury's  Orbit.  —  Leverrier, 
in  1859,  from  a  discussion  of  all  the  observed  transits  of  Mercury, 
found  that  the  perihelion  of  its  orbit  has  a  movement  of  nearly  38" 
a  century  over  and  above  what  can  be  accounted  for  by  the  action  of 
the  known  planets,  and  he  calculated  that  it  could  be  explained  by 
the  attraction  of  a  planet,  or  ring  of  small  planets,  revolving  inside 
this  orbit  nearly  in  its  plane,  with  a  mass  about  half  as  great  as  that 
of  Mercury  itself. 

It  could  also  be  explained  on  the  hypothesis  that  the  force  of  gravitation 
instead  of  varying  strictly  as  —  varies  as  _  g    n  ,  where  n  is  an  extremely 

small  quantity ;  also  on  the  hypothesis  that  the  law  of  attraction  is  not 
exactly  the  same  for  bodies  in  motion  as  at  rest,  being  slightly  less  in  the 
former  case,  —  the  so-called  electro-dynamic  theory  of  gravitation.  But 
Newcomb  finds  that  while  the  motions  of  Mercury's  perihelion  may  be 
explained  in  these  various  ways,  the  motion  of  his  node  (and  that  of  Venus 
also)  appears  to  be  inconsistent  with  the  existence  of  such  a  planetary  ring, 
and  the  subject  is  by  no  means  cleared  up  as  yet. 

603.  Dr.  Lescarbault's  Observation:  Vulcan.  —  A  certain  country 
physician,  living  some  eighty  miles  from  Paris,  Dr.  Lescarbault,  on  the  pub- 
lication of  Leverrier's  result,  announced  that  he  had  actually  seen  this  planet 
crossing  the  sun  nine  months  before,  on  the  26th  of  March  of  that  year, 
1859.    He  was  visited  by  Leverrier,  who  became  satisfied  of  the  genuineness 


374  LNTRA-MERCUKIAL   PLANETS. 

of  his  observations,  and  the  doctor  was  duly  congratulated  and  honored  as 
the  discoverer  of  "Vulcan,"  which  name  was  assigned  to  the  supposed  new 
planet.  An  interesting  account  of  the  matter  may  be  found  in  Chambers' 
"Descriptive  Astronomy";  and  in  many  of  the  works  published  from  twenty 
to  twenty-five  years  ago,  as  well  as  in  some  more  recent  ones,  "  Vulcan "  is 
assigned  a  place  in  the  solar  system,  with  a  distance  of  about  13  000000 
miles  and  a  period  of  19  -f-  days.  Lescarbault  described  it  as  having  an 
apparent  diameter  of  about  7",  which  would  make  it  over  2500  miles  in 
diameter. 

604.  Nevertheless,  it   is    nearly  certain  that  Vulcan  does  not  exist. 
There  are  various  opinions  which  we  need  not  here  discuss  as  to  the  ex- 
planation of  this  pseudo-discovery.      But  the  planet,  if  real,  ought  since 
1859  to  have  been  visible  on  the1  sun's  face  at  certain  definite  times  which 
Leverrier  calculated  and  published ;    and  it  has  never  been  seen,  though 
very  carefully  looked  for.     Small,  round,  dark  objects  have  from  time  to 
time  been  indeed  reported  on  the  sun's  disc,  which  in  the  opinion  of  the 
observers  at  the  time  were  not  sun  spots  ;   but  most  of  these  observations 
were  made  by  amateurs  with  comparatively  little  experience,  with  small 
telescopes,  and  with  no  measuring  apparatus  by  which  they  could  certainly 
determine  whether  or  not  the  spot  seen  moved  like  a  planet.     In  most  of 
these  cases  photographs  or  simultaneous  observations  made  elsewhere  by 
astronomers  of  established  reputation,  and  having  adequate  apparatus,  have 
proved  that  the  problematical  "dots"  were  really  nothing  but  ordinary 
small  sun  spots,  and  the  probability  is  that  the  same  explanation  applies  to 
the  rest. 

605.  Eclipse  Observations.  —  A  planet  large  enough  to  be  seen 
distinctly  on  the  sun  by  a  2£-inch  telescope,  such  as  Lescarbault 
used,  would  be  a  conspicuous  object  at  the  time  of  a  solar  eclipse, 
and  most  careful  search  has  been  made  for  the  planet  on  such  occa- 
sions ;  but  so  far,  although  stars  of  the  third  and  fourth  magnitudes, 
and  even  of  the  fifth,  have  been  clearly  seen  by  the  observers  within 
a  few  degrees  of  the  eclipsed  sun,  no  planet  has  been  found. 

One  apparent  exception  occurred  in  1878.  During  the  eclipse  of  that 
year,  Professor  Watson  observed  two  starlike  objects  (of  the  fourth  magni- 
tude), which  he  thought  at  the  time  could  not  be  identified  with  any  known 
stars  consistently  with  his  observations.  Mr.  Swift,  also,  at  the  same  eclipse, 
reported  the  observations  of  two  bright  points  very  near  the  sun ;  but  these 
from  his  statement  could  not  (both)  have  been  identical  with  Watson's  stars. 
Later  investigations  of  Dr.  Peters  have  shown  that  the  assumption  of  a 
very  small  and  very  likely  error  in  Professor  Watson's  circle-readings 
(which  were  got  in  a  very  ingenious,  but  rather  rough  way,  without  the 
use  of  graduations)  would  enable  his  stars  to  be  identified  with  0  and  £ 
Cancri,  and  it  is  almost  certain  that  these  were  the  stars  he  saw.  Mr. 


THE   ZODIACAL   LIGHT.  375 

Swift's  observations  remain  unexplained.  With  this  exception,  the  eclipse 
observations  all  give  negative  results,  and  astronomers  generally  are  now  dis- 
posed to  consider  the  "Vulcan  question"  as  settled  definitely  and  adversely. 

606.  At  the  same  time  it  is  extremely  probable  that  there  are  a 
number,  and  perhaps  a  very  great  number,  of  intra-Mercurial  aste- 
roids.    A  body  two  hundred  miles  in  diameter  near  the  sun  would 
have  an  angular  diameter  of  only  about  -J-",  as  seen  from  the  earth, 
and  would  not  be  easily  visible  on  the  sun's  disc,  except  with  very 
large  telescopes.     It  would  not  be  at  all  likely  to  be  picked  up  acci- 
dentally.    Objects  with  a  diameter  of  not  more  than  forty  or  fifty 
miles  would  be  almost  sure  to  escape  observation,  either  at  a  transit 
or  during  a  solar  eclipse  unless,  possibly,  by  photography. 

607.  Zodiacal  Light.  —  This  is  a  faint,  pyramidal  haze  of  light  that 
extends  from  the  sun  along  the  ecliptic.     In  the  evening  it  is  best  seen  in 
February,  March,  and  April,  because  the  portion  of  the  ecliptic  which  lies 
east  of  the  sun's  place  is  then  most  nearly  perpendicular  to  the  western 
horizon.     During  the  autumnal  months  the  zodiacal  light  is  best  seen  in  the 
morning  sky  for  a  similar  reason.     In  our  latitudes  it  can  seldom  be  traced 
more  than  90°  or  100°  from  the  sun  ;    but  at  high  elevations  within  the 
tropics  it  is  said  to  extend  entirely  across  the  sky,  forming  a  complete  ring, 
and  there  is  said  to  be  in  it  at  the  point  exactly  opposite  to  the  sun  a  patch 
a  few  degrees  in  diameter  of  slightly  brighter  luminosity,  called  the  "  Gegen- 
schein  "  or  "  counter-glow." 

The  portions  of  this  object  near  the  sun  are  reasonably  bright,  and  even 
conspicuous  at  the  proper  seasons  of  the  year ;  but  the  more  distant  portions 
in  the  neighborhood  of  the  "  counter-glow  "  are  so  extremely  faint  that  it  is 
only  possible  to  observe  them  at  a  distance  from  cities  and  large  towns,  in 
places  where  the  air  is  free  from  smoke,  and  where  the  darkness  of  the  sky 
is  not  affected  by  the  general  illumination  due  to  gas  and  electric  lights. 

Its  spectrum  has  been  observed  by  Wright  of  New  Haven  and  others, 
and  appears  to  be  continuous,  showing  no  bright  lines  —  it  is  too  faint  to 
show  the  dark  lines  of  the  solar  spectrum  if  they  are  really  present,  as  is 
very  probable,  since  the  light  appears  to  be  partially  polarized  as  if  reflected 
from  minute  particles. 

It  has  often  been  stated  that  the  spectrum  of  the  zodiacal  light  shows  the 
same  bright  line  which  characterizes  that  of  the  Aurora  Borealis :  this  is  a 
mistake. 

608.  The  cause  of  the  phenomenon  is  not  certainly  known,  but  at  pres- 
ent the  theory  most  generally  accepted  attributes  it  to  sunlight  reflected  by 
myriads  of  small  meteoric  bodies  which  are  revolving  around  the  sun  nearly  in 
the  plane  of  the  ecliptic,  forming  a  thin,  flat  sheet  like  one  of  Saturn's  rings, 
and  extending  far  beyond  the  orbit  of  the  earth.    It  may  be  that  the  denser 


376  THE   ZODIACAL   LIGHT. 

portion  of  this  meteoric  ring  within  the  orbit  of  Mercury  is  the  cause  of  the 
motion  of  the  perihelion  of  that  planet  which  Leverrier  detected  ;  it  is  for 
this  reason  that  we  deal  with  the  subject  here  rather  than  in  connection  with 
meteors.  While  this  theory,  however,  is  at  present  more  generally  accepted 
than  any  other,  it  cannot  be  said  to  be  established.  Some  are  disposed  to 
consider  the  zodiacal  light  as  a  mere  extension  of  the  sun's  corona,  whatever 
that  may  be. 


EXERCISES  ON  CHAPTER  XV. 

1.  On  May  2,  1896,  the  apparent  semi-diameter  of  Jupiter  was  17".75,  its 
distance  from  the  earth  being  at  that  time  5.431  Astron.  units.  Required 
the  planet's  diameter  compared  with  that  of  the  earth. 

Remember  that  the  solar  parallax,  8".80,  is  the  same  as  the  earth's  semi-diameter  seen 
from  distance  unity. 

17.75 
Ans.    0  Qr.   X  5.431,  or  10.95  times  diameter  of  the  earth. 

o.oO 

*'  2.  Assuming  the  preceding  measure  as  exact,  what  ought  to  be  the 
apparent  semi-diameter  of  the  planet  when  at  a  distance  of  4.25  ? 

Ans.   22".68. 

"  3.  What  must  be  the  mass  of  the  earth  to  make  the  moon  revolve  around 
it  with  the  same  period  as  now,  but  at  twice  its  present  distance  ?  (See 
Art.  537.) 

Make  E  the  present  mass  of  the  earth,  and  E'  the  required  mass,  and  apply  the  equation 
given  in  Art.  537. 

Ans.    E'^&E. 

"  4.  How  much  must  the  mass  of  the  earth  be  increased  to  make  the 
moon,  at  its  present  distance,  revolve  in  two  days  ? 

(97  39  \  2 
^p  J  ,  =  186.6  E. 

5.  What  reduction  of  the  earth's  mass,  suddenly  produced,  would  release 
the  moon,  i.e.,  transform  her  orbit  into  a  parabola  or  hyperbola  ? 

Ans.    Any  reduction  exceeding  50  per  cent. 

6.  At  what  rate  does  the  elongation  of  Venus  from  the  sun  change  at  or 
near  the  time  of  superior  conjunction?     (See  Art.  564.) 

38".81  hourly,  or 


J  38".81  hourly,  ( 
Ans'  \  15'  31". 4  daily. 


EXERCISES.  377 

7.    At  what  rate  does  Venus  appear  to  cross  the  sun's  disc  during  a 
transit?     (See  Art.  575).  4n*.   241"  per  hour. 


8.   At  what  rates  does  Mars  advance  when  at  conjunction,  and  retrograde 
at  opposition  ? 

(  At  conjunction,  42'.43  daily  advance. 
'  \  At  opposition,  2F.43  daily  regression. 


NOTE  TO  ART.  573. 

The  observation  of  Professor  Lyman  was  repeated  by  Russell  at  Princeton  in 
December,  1898. 

His?  investigations  indicate  that  refraction  plays  a  comparatively  small  part 
in  the  formation  of  the  luminous  ring,  which  appears  to  be  mainly  due  to 
diffusion  of  light,  like  that  which  produces  our  own  twilight.  If  due  to  refrac- 
tion, the  ring  should  be  widest  and  brightest  on  the  side  of  the  planet  farthest 
from  the  sun,  while  the  reverse  is  the  case. 

Nor  do  they  furnish  any  evidence  that  the  planet's  atmosphere  is  more  dense 
or  extensive  than  the  earth's,  but  rather  the  contrary  ;  as  might  be  expected 
considering  her  smaller  mass,  and  probably  higher  temperature. 

NOTE  TO  ART.  595. 

EROS.  The  planet  Eros,  (433),  is,  in  many  ways,  the  most  interesting  of  the 
Asteroid  group  thus  far  known.  It  was  discovered,  photographically,  by  Witt 
of  Berlin,  in  August,  1898,  and  at  once  attracted  attention  by  its  rapid  motion. 
Its  mean  distance  from  the  sun  is  not  quite  135,480000  miles,  —  less  than  that 
of  Mars,  and  its  period  is  643  days.  The  eccentricity  of  its  orbit  is  0.223,  so 
that  its  aphelion  distance  is  165,670000  miles,  and  its  perihelion  distance, 
105,290000,  exceeding  the  mean  distance  of  the  earth  by  only  about  12  million 
miles.  The  orbits,  however,  are  so  situated  that  it  cannot  come  nearer  than 
13|  million.  Still  this  is  only  a  little  more  than  half  the  least  distance  of 
Venus,  and  observations  of  Eros,  made  at  such  a  time  of  closest  approach,  will 
furnish  by  far  the  most  precise  means  known  for  determining  the  solar  parallax. 
Unfortunately  these  close  oppositions  are  rare  :  one  occurred  in  189^  (before  the 
planet  was  discovered),  and  another  such  opportunity  will  not  occur  until  the 
year  1931  ;  though  in  1901  the  conditions  were  far  better  than  usual,  and  better 
than  they  will  be  again  until  1931  and  1938. 

The  inclination  of  the  planet's  orbit  is  nearly  11°,  and  when  nearest  the 
earth,  as  in  1894,  it  goes  into  circumpolar  declinations,  and  at  the  time  of  oppo- 
sition it  moves  almost  directly  south,  its  motion  in  right-ascension  being  very 
slight. 

The  planet  is  extremely  small,  probably  not  exceeding  20  miles  in  diameter, 
and  seldom  visible  except  in  the  largest  telescopes,  though  when  nearest  it  is 
just  possible  that  it  may  reach  the  naked-eye  limit.  As  already  stated  it  ex- 
hibits at  times  regular  variations  of  brightness  apparently  indicating  a  rotation- 
period  of  5h  16m,  though  a  different  explanation  is  possible. 


378  THE   MAJOR   PLANETS. 


CHAPTER   XVI. 

THE   PLANETS    CONTINUED.  —  THE  MAJOR   PLANETS :    JUPITER, 
SATURN,   URANUS,   AND   NEPTUNE. 

JUPITER. 

609.  While  this  planet  is  not  so  brilliant  as  Venus  at  her  best, 
it  stands  next  to  her  in  this  respect,  being  on  the  average  about  five 
times  brighter  than  Sirius,  the  brightest  of  the  fixed  stars.     Jupiter, 
moreover,  being  a  "superior"  planet,  is  not  confined,  like  Venus, 
to  the  neighborhood  of  the  sun,  but  at  the  time  of  opposition  is  the 
chief  ornament  of  the  midnight  sky. 

610.  Orbit. — The  orbit  presents  no  marked  peculiarities.     The 
mean  distance  of  the  planet  from  the  sun  is  483,000000  miles.     The 
eccentricity  of  the  orbit  being  nearly  ^  (0.04825)  ;  the  greatest  and 
least   distances  vary  by  about    21,000000  miles  each  way,  making 
the  planet's  greatest  and    least  distances  from  the  sun  504,000000 
and  462,000000   miles   respectively.     The   average   distance  of  the 
planet  from  the  earth  at  opposition  is  390,000000,  while  at  conjunc- 
tion it  is   576,000000    miles.     The  minimum  opposition  distance  is 
only  369,000000,  which  is  obtained  when  the  opposition  occurs  about 
October  6,  Jupiter  being  in  perihelion  when  its  heliocentric  longi- 
tude is  about  12°.     At  an  aphelion  opposition  (in  April)  the  distance 
is  42,000000  miles  greater;  that  is,  411,000000. 

The  rela-tive  brightness  of  Jupiter  at  an  average  conjunction  and 
at  the  nearest  and  most  remote  oppositions  is  respectively  as  the 
4-  numbers  10,  27,  and  18.  The  average  brightness  at  opposition  is, 
therefore,  more  than  double  that  at  conjunction;  and  at  an  October 
opposition  the  planet  is  fifty  per  cent  brighter  than  at  an  April  one. 
The  differences  are  considerable,  but  far  less  important  than  in  the 
case  of  Mars,  Venus,  and  Mercury. 

The  inclination  of  the  orbit  to  the  ecliptic  is  small,  — only  1°  19'. 

611.  Period.  —  The  sidereal  period  is  11.86  years,  and  the  synodic 
is  399  days  (a  number  easily  remembered) ,  a  little  more  than  a  year 
and  a  month.     The  planet's  orbital  velocity  is  about  eight  miles  a 
second. 


JUPITER.  379 

612.  Dimensions.  —  The  planet's  apparent  diameter  varies  from 
50"  at  an  October  opposition   (or  451"  at  an  April  one)   to  32"  at 
conjunction.     The  form,  however,  of  the  planet's  disc  is  not  truly 
circular,  the  polar  diameter  being  about  ^  part  less  than  the  equa- 
torial, so  that  the  eye  notices  the  oval  form  at  once.     The  equa- 
torial diameter  in  miles  is  90190,  the  polar  being  84570.     Its  mean  1 
diameter,  therefore,   is  88300,  —  almost  eleven  times  that  of  the 
earth. 

This  makes  its  surface  119  times,  and  its  volume  1300  times,  that 
of  the  earth.  It  is  by  far  the  largest  of  the  planets  in  the  system  ; 
in  fact,  whether  we  regard  its  bulk  or  its  mass,  larger  than  all  the  rest 
put  together. 

613.  Mass,  Density,  etc.  —  Its  mass  is   very  accurately  known, 
both  by  the  motions  of  its  satellites,  and  the  perturbations  of  the 
asteroids.    It  is  T^J7.?  of  the  sun's  mass,  or  very  nearly  318  times 
that  of  the  earth.    Comparing  this  with  its  volume,  we  find  its  density 
0.24,  less  than  \  the  density  of  the  earth,  and  almost  precisely  the 
same  as  that  of  the  sun.     Its  mean  superficial  gravity  comes  out  2.64 
times  that  of  the  earth  ;    that  is,  a  body  on  Jupiter  would  weigh  2f 
times  as  much  as  upon  the  surface  of  the  earth  ;    but  on  account  of 
the  rapid  rotation  of  the  planet  and  its  ellipticity  there  is  a  very  con- 
siderable difference  between  the  force  of  gravity  at  the  equator  and 
at  the  pole,  amounting  to  -J-  of  the  equatorial  gravity.     (On  the  earth 
the  difference  is  only 


614.  Phases  and  Albedo,  —  Its  orbit  is  so  much  larger  than  that 
of  the  earth  that  the  planet  shows  no  sensible  phases,  even  at  quadra- 
ture, though  at  that  time  the  edge  farthest  from  the  sun  shows  a 
slight  darkening. 

The  reflecting  power,  or  Albedo,  of  the  planet's  surface  is  very 
high,  —  0.62  according  to  Zollner,  that  of  white  paper  being  only 
0.78.  The  centre  of  the  disc  of  this  planet  (and  the  same  is  also 
true  of  Saturn)  is  considerably  brighter  than  the  limb  —  just  the 
reverse,  as  will  be  remembered,  from  the  condition  of  things  upon 
the  moon,  and  upon  Mars,  Venus,  and  Mercury.  This  peculiarity 
of  a  darkened  limb,  in  which  Jupiter  resembles  the  sun,  has  sug- 

1  The  mean  diameter  of  an  oblate  spheroid  is  _£+_,  not  2L±__.     Of  the  three 

axes  of  symmetry  which  cross  at  right  angles  at  the  planet's  centre,  one  is  the 
axis  of  rotation,  and  both  the  others  are  equatorial. 


380  JUPITEE. 

gested  the  idea  that  it  is  to  some  extent  self-luminous.  This,  how- 
ever, is  not  a  necessary  consequence,  as  a  nearly  transparent  atmos- 
phere overlying  a  uniformly  reflecting  surface  would  produce  the 
same  effect. 

The  light  which  the  planet  emits,  if  it  emits  any,  must  be  very 
feeble  as  compared  with  sunlight,  since  the  satellites,  when  they  are 
eclipsed  by  entering  the  shadow,  become  totally  invisible. 

615.  Axial  Rotation.  —  The  planet  rotates  on  its  axis  in  about 
9h  55m.     The  time  can  be  given  only  approximately,  not  because 
it  is  difficult  to  find  and  observe  distinct  markings  on  the  planet's 
disc,  but  simply  because  different  results  are  obtained  from  differ- 
ent spots,  according  to  their  nature   and   their  distance   from   the 
planet's  equator.     Speaking  generally,  spots  near  the  equator  indi- 
cate a  shorter  day  than  those  in  higher  latitudes,  and  certain  small, 
sharply  denned,  bright,  white  spots,  such  as  are  often  seen,  give  a 
quicker  rotation  than  the  dark  markings  in  the  same  latitude. 

According  to  Williams  there  are  at  least  nine  "  belts "  of  atmospheric 
current  on  Jupiter,  clearly  distinct  from  each  other ;  the  swiftest,  at  the 
equator,  has  a  rotation-period  of  only  9h  50m  20s,  while  that  of  the  slowest 
is  9h  56m.  The  great  red  spot  has  given  values  ranging  from  9h  55m  34S.9 
(in  1879)  to  9&  55™  4(K7  (in  1886),  and  9h  55^  41«.5  (in  1906).  The  in- 
crease has  been  unmistakable,  and  is  not  due  to  any  uncertainty  in  the 
observations. 

616.  The  Axis  of  Rotation  and  the  Seasons.  —The  plane  of  the 
equator  is  inclined  only  3°  to  that  of  the  orbit,  so  that  as  far  as  the 
sun  is   concerned   there   can  be  no  seasons.     The   heat   and   light 
received  from    the    sun  by  Jupiter  are,  however,  only  about  ^V  as 
intense   as   the   solar  radiation  at  the  earth,  its  distance  being  5.2 
times  as  great. 

617.  Telescopic   Appearance. — Even   in   a   small   telescope   the 
planet  is  a  beautiful  object.     When  near  opposition  a  magnifying 
power  of  only  40  makes  its  apparent  size  equal  to  that  of  the  full 
moon  (though,  as  remarked  in  connection  with  Venus,   no  novice 
would  receive  that  impression) ,  and  with  a  telescope  of  8  or  10  inches 
aperture,  and  with  a  magnifying  power  of  300  or  400,  the  disc  is 
covered  with  an  infinite  variety  of  beautiful  and  interesting  details 
which  rapidly  shift  under  the  observer's  eye   in  consequence   of  the 
planet's  swift  rotation.    The  picture  is  rich  in  color,  also,  browns  and 
reds   predominating,   in  contrast   with   olive-greens   and   occasional 


THE    GREAT   RED    SPOT. 


381 


purples ;  but  to  bring  out  the  colors  well  and  clearly  requires  large 
instruments.  For  the  most  part  the  markings  are  arranged  in  streaks 
more  or  less  parallel  to  the  planet's  equator,  as  shown  by  Fig.  178. 
With  a  small  telescope  the  markings  usually  reduce  to  two  dark  and 
comparatively  well-defined  belts,  one  on  each  side  of  the  equator, 
occupying  about  the  same  regions  of  latitude  that  the  trade-wind 
zones  do  upon  the  earth ;  and  very  likely  in  Jupiter's  case  similar 
aerial  currents  have  something  to  do  with  the  appearance,  though 
upon  Jupiter,  as  has  been  already  said,  the  solar  heat  is  a  coinpara- 


FIG.  178.  — Telescopic  Views  of  Jupiter. 


tively  unimportant  factor.  The  markings  upon  the  planet  are  almost, 
if  not  entirely,  atmospheric,  as  is  proved  by  the  manner  in  which 
they  change  their  shapes  and  relative  positions.  They  are  cloud 
forms.  It  is  hardly  probable  that  we  ever  see  anything  upon  the 
solid  surface  of  the  planet  underneath,  nor  is  it  even  certain  that 
the  planet  has  anything  solid  about  it.  In  Fig.  178,  the  upper 
left-hand  figure  is  from  a  drawing  by  Trouvelot  made  in  February, 
1872  ;  the  second  is  by  Vogel  in  1880.  The  small  one  below  repre- 
sents the  planet  as  seen  in  a  small  telescope. 

618.    The  Great  Red  Spot. — While  most  of  the  markings  on  the 
planet  are  evanescent,  it  is  not  so  with  all.     There  are  some  which 


382 


JUPITER. 


are  at  least  "  sub-permanent/'  and  continue  for  years,  not  without 
change  indeed,  but  with  only  slight  changes.  The  "  great  red  spot " 
is  the  most  remarkable  instance  so  far.  It  seems  to  have  been  first 
observed  by  Prof.  C.  W.  Pritchett  of  Glasgow,  Missouri,  in  July, 
1878,  as  a  pale,  pinkish,  oval  spot  some  13"  in  length  by  3"  in 
width  (30,000  miles  by  7000).  Within  a  few  months  it  had  been 
noticed  by  a  considerable  number  of  other  observers,  though  at  first 
it  did  not  attract  any  special  attention,  since  no  one  thought  of  it  as 
likely  to  be  permanent.  The  next  year,  however,  it  was  by  far  the 
most  conspicuous  object  on  the  planet.  It  was  of  a  clear,  strong 
brick-red  color,  with  a  length  fully  one-third  the  diameter  of  the 
planet  and  a  width  about  one-fourth  of  its  length. 

For  two  or  three  years  it  remained  without  much  change ;  in  1882-83 
it  gradually  faded  out ;  in  1885  it  had  become  a  pinkish  oval  ring,  the 
central  part  being  apparently  occupied  with  a  white  cloud.  In  1886  it  was 
again  a  little  stronger  in  color,  and  the  same  in  1887, —  an  object  not  diffi- 

l  2 


3  4 

FIG.  179.  — Jupiter's  "  Bed  Spot."    From  Drawings  by  Mr.  Denning.    1880-85. 

cult  to  see  with  a  large  telescope,  but  the  merest  ghost  of  what  it  was  in 
1880.  It  still  persists  (1908),  though  extremely  faint,  having  shortened 
up  a  little  and  lost  its  pointed  ends.  It  lies  at  the  southern  edge  of  the 
southern  equatorial  belt,  in  latitude  about  30°,  and  for  some  reason  the  belt 


TEMPERATURE   AND   PHYSICAL   CONSTITUTION.  383 

seems  to  be  "  notched  out  "  for  it.  Even  when  the  spot  was  palest  its  place 
was  always  evident  at  once  from  the  indentation  in  the  outline  of  the  belt. 

It  lies  in  an  atmospheric  belt  which  has  a  rotation-period  22  seconds 
shorter  than  its  own,  so  that,  to  quote  the  expression  of  Williams,  it  "emerges 
like  an  island  in  a  river,"  the  current  drifting  past  it  at  a  rate  of  12  or  15 
miles  an  hour. 

Such  phenomena  suggest  abundant  matter  for  speculation.  It  must  suffice 
to  say  that  no  satisfactory  explanation  of  the  phenomena  has  yet  been  pre- 
sented. The  unquestionable  fact  before  mentioned  (Art.  615),  that  the 
time  of  rotation  of  the  spot  has  changed  by  more  than  6s,  greatly  complicates 
the  subject.  Fig.  179,  from  the  drawings  of  Mr.  Denning,  represents  the 
appearance  of  the  spot  at  four  different  dates ;  viz.,  1,  1880,  Nov.  19  ;  2, 
1882,  Oct.  30 ;  3,  1884,  Feb.  6 ;  4,  1885,  Feb.  25. 

619.  Temperature  and  Physical  Constitution.  —  The  rapidity  of 
the  changes  upon  the  visible  surface  implies  the  expenditure  of  a 
considerable  amount  of  heat,  and  since  the  heat  received  from  the 
sun  is  too  small  to  account  for  the  phenomena  which  we  see,  Zollner, 
thirty  years  ago,  following  the  suggestions  of   Buffon  and  Kant, 
practically  demonstrated  that  it  must  come  from  within  the  planet, 
and  that  in  all  probability  Jupiter  is  at  a  temperature  not  much 
short  of  incandescence,  —  hardly  yet  solidified  to  any  considerable 
extent.     Most   astronomers   suppose  the   visible   features   on   the 
planet's  surface  to  be  purely  atmospheric,  but  Hough  considers  that 
we  see  the  pasty,  semi-liquid  surface  of  the  globe  itself. 

620.  Atmosphere.  —  As  to  the  composition  of  the  planet's  atmos- 
phere, the  spectroscope  gives  us  rather  surprisingly  little  informa- 
tion.    We  get  from  the  planet  a  good  solar  spectrum  with  the  solar 
lines  well  marked,  but  there  are  no  well-defined  absorption  bands 
due  to  the  action  of  the  planet's  atmosphere.     There  are,  however, 
some  shadings  in  the  lower  red  portion  of  the  spectrum  that  are 
probably  thus  caused.     The  light,  for  the  most  part,  seems  to  come 
from  the  upper  surface  of  the  planet's  envelope  of  clouds  without 
having  penetrated  to  any  depth. 

Spectroscopic  observations  upon  the  relative  shift  of  the  dark  lines  in  the 
spectrum  at  the  eastern  and  western  limbs,  give  a  very  fair  determination 
of  its  rotation-period  (by  Doppler's  principle). 

621.  Satellite  System.  —  Jupiter  has  seven l  satellites,  —  four  of 
them  the  first  heavenly  bodies  ever  discovered  —  the  first  revelation 
of  Galileo's  telescope.     His  earliest  observation  of  them  was  on 
Jan.  7,  1610,  and  in  a  very  few  weeks  he  had  ascertained  their  true 
character,  and  determined  their  periods  with  an  accuracy  which  is 

1  For  the  sixth  and  seventh  satellites,  see  note  on  page  406. 


384  JUPITER. 

surprising.  The  number  of  the  heavenly  bodies  was  now  no  longer 
|  ^sevevtj  and  the  discovery  excited  among  churchmen  and  schoolmen 
a  great  deal  of  angry  incredulity  and  vituperation.  Galileo  called 
them  "the  Medicean  stars." 

These  four  are  usually  known  as  the  first,  second,  etc.,  in  the 
order  of  distance  from  the  primary,  but  they  also  have  names  which 
are  sometimes  used;  viz.j  lo,  Europa,  Ganymede,  and  Callisto. 
Their  relative  distances  range  between  262000  and  1  169000  miles, 
being  very  approximately  6,  9,  15,  and  26  radii  of  the  planet.  Their 
sidereal  periods  range  between  ld  18£h  and  16d  16£h. 

The  fifth  satellite  was  discovered  by  Mr.  Barnard  at  the  Lick 
Observatory  in  September,  1892.  It  is  extremely  small,  and  so  near  the 
planet  that  it  is  exceedingly  difficult  to  see,  and  quite  out  of  reach  of 
any  telescopes  less  than  18  or  20  inches  in  aperture.  Its  distance  from 
the  planet's  centre  is  about  112500  miles,  and  its  period  llh  57.  4m. 

The  orbits  of  all  five  of  these  satellites  are  almost  circular,  and  lie 
very  nearly  in  the  plane  of  Jupiter's  equator. 

The  satellites  slightly  disturb  each  other's  motions,  and  from  these 
disturbances  their  masses  can  be  ascertained  in  terms  of  the  planet's  mass. 
The  third,  which  is  much  the  largest,  has  a  mass  of  about  TT^OQ  °^  the 
planet's,  a  little  more  than  double  the  mass  of  our  own  moon.  The  mass 
of  the  first  satellite  appears  to  be  a  little  less  than  ^  as  much.  The  second 
is  somewhat  larger  than  the  first,  and  the  fourth  is  about  half  as  large  as  the 
third  ;  i.e.,  it  has  about  the  mass  of  our  own  moon.  The  densities  of  the 
first  and  fourth  appear  to  be  not  very  different  from  that  of  the  planet  itself, 
while  the  densities  of  the  second  and  third  are  considerably  greater. 

622,  Relation  between  Mean  Motions  and  Longitudes  of  the 
Satellites.  —  In  consequence  of  their  mutual  interaction  a  curious  relation 
(discovered  by  La  Place)  exists  between  the  mean  motions  of  the  first  three 
satellites.  The  mean  motion  is  of  course  360°  divided  by  T  (T  being  the 
satellite's  period).  It  appears  that  the  mean  motion  of  the  first  plus  twice 
the  mean  motion  of  the  third  equals  three  times  that  of  the  second,  or 

i  JL  JL  -  JL 

T         T         T' 
*1          JS          22 

A  similar  relation  holds  for  their  longitudes  : 


so  that  they  cannot  all  three  come  into  opposition  or  conjunction  with  the 
sun  at  once.  These  relations  are  permanently  maintained  by  their  mutual 
attractions  :  exactly  in  the  long  run,  though  there  are  slight  perturbations  pro- 
duced by  the  fourth  satellite  which  disturb  the  arrangement  slightly  for  short 
periods.  The  fourth  and  fifth  satellites  do  not  come  into  the  arrangement. 


JUPITER'S  SATELLITES.  385 

623.  Diameters,  etc.  —  The  diameter  of  the  first  satellite  is  a  little 
more  than  2400  miles ;  the  second  is  almost  exactly  the  size  of  our  own 
moon,  i.e.,  between  2100  and  2200  miles ;  and  the  third  and  fourth  have  diam- 
eters, respectively,  of  3600  and  3000  miles,  the  third,  Ganymede,  being  much 
larger  than  either  of  his  sisters.     When  Jupiter  is  in  opposition,  the  fourth 
satellite  is  sometimes  nearly  10^'  away  from  the  planet,  or  £  of  the  moon's 
diameter ;  and  in  very  clear  air  can  be  seen  by  a  sharp  eye  without  telescopic 
aid.     The  third,  though  much  larger,  never  goes  more  than  6'  from  the 
planet,  and  it  is  perhaps  doubtful  whether  it  is  ever  seen  with  the  naked  eye, 
unless  when  the  fourth  happens  to  be  close  beside  it.     A  good  opera-glass 
will  easily  show  them  all  as  minute  points  of  light.    The  fifth  (new)  satellite 
can  hardly  exceed  100  miles  in  diameter. 

624.  Brightness.  —  Since  the  sunlight  of  Jupiter  is  only  -fa  as  intense 
as  ours,  the  moonlight  made  by  the  satellites  is  decidedly  inferior  to  our  own, 
although  their  reflective  power  appears  to  be  higher  than  that  of  the  lunar 
surface.     They  differ  among  themselves  considerably  in  this  respect.     The 
fourth  satellite  is  of  an  especially  dark  complexion.      The  others,  under 
similar  circumstances,  show  light  or  dark   according  as  they  have  a  dark 
or  light  portion  of  the  planet  for  a  background.     Even  the  fourth,  when 
crossing  the  disc,  is  always  seen  bright  while  very  near  the  planet's  limb. 

625.  Markings  upon  the  Satellites.  —  The  satellites  show  sensible 
discs  when  viewed  with  a  large  telescope,  and  all  of  them  but  the  second 
sometimes  show  dark  markings  upon  the  surface.    These  markings,  however, 
are  only  visible  under  the  most  favorable  circumstances,  and  it  has  not  been 
possible  to  determine  whether  they  are  atmospheric  or  really  geographical, 
nor  to  deduce  from  them  with  certainty  the  satellites'  periods  of  rotation.1 
W.  Pickering  has  also  reported  certain  periodical  changes  of  form  in  the  first 
and  second  satellites,  as  if  they  were  whirling  clouds  or  meteoric  swarms,  and 
not  solid  bodies.     But  his  observations  require  confirmation. 

626.  Variability.  —  Galileo  noticed  variations  in  the  brightness  of  the 
satellites  at  different  times,  and  subsequent  observers  have  confirmed  his 
result.      In  the  case  of  the  fourth  satellite  there  seems  to  be  a  regular 
variation  depending  upon  the  place  of  the  satellite  in  its  orbit,  and  suggest- 
ing that  in  its  axial  rotation  it  behaves  like  our  own  moon,  keeping  always 
the  same  side  next  its  primary.    In  addition  it  shows  other  irregular  changes 
in  its  luminosity:   so  also  do  the  other  satellites  according  to  nearly  all 
authorities,  though  it  is  singular  that  one  or  two  of  the  best  observers  do 
not  find  any  such  irregularity  indicated  by  their  instrumental  photometric 
observations. 

1  Mr.  Douglas  of  the  Flagstaff  Observatory  reports  in  1897  observations  of  the 
markings  showing  that  the  third  and  fourth  satellites  rotate  like  the  moon,  in 
periods  sensibly  identical  with  their  orbital  revolutions,  confirming  the  earlier 
conclusion  referred  to  in  Article  626. 


386 


JUPITER. 


627.  Eclipses  and  Transits.  —  The  satellites'  orbits  are  so  nearly 
in  the  plane  of  the  planet's  orbit  that,  excepting  the  fourth,  they  all 
pass  through  the  shadow  of  the  planet,  and  suffer  eclipse  at  every 
revolution.  At  conjunction,  also,  they  cast  their  shadows  upon  the 
planet,  and  these  shadows  can  easily  be  seen  in  the  telescope  as 
black  dots  on  the  planet's  disc,  the  satellites  themselves,  which  cross 
the  disc  about  the  same  time,  being  much  more  difficult  to  observe. 
The  fourth  satellite  escapes  eclipse  when  Jupiter  is  far  from  the 
node  of  its  orbit.  Thus,  during  1894  and  the  first  three  months  of 
1895,  there  were  no  eclipses  of  Callisto  at  all. 

Exactly  at  opposition  or  conjunction  the  planet's  shadow  lies 
straight  behind  it  out  of  our  sight,  so  that  we  cannot  at  that  time 


FIG.  180.  —  Eclipses  of  Jupiter's  Satellites,  at  Western  Elongation. 

observe  the  eclipses,  but  only  their  transits  across  the  disc.     Before 
and  after  these  times  the  shadow  lies  one  side  of  the  planet. 

When  the  planet  is  at  quadrature  and  the  condition  of  things  is  as 
represented  in  Fig.  180  (which  is  drawn  to  scale),  the  shadow  projects 
so  far  to  one  side  of  the  planet  that  the  whole  eclipse  of  all  the  satel- 
lites, except  the  first,  takes  place  clear  of  the  planet's  disc,  —  both 
the  disappearance  and  reappearance  of  the  satellite  being  visible. 


"Equation  of  Light."  —  The  most  important  use  that  has 
been  made  of  these  eclipses  has  been  to  ascertain  the  time  required 
by  light  in  traversing  the  distance  between  us  and  the  sun,  the  so- 


THE  EQUATION  OF  LIGHT. 


387 


called  "  equation  of  light."  It  was  in  1675  that  Roemer,  the  Danish 
astronomer  (the  inventor  of  the  transit  instrument,  meridian  circle, 
and  prime  vertical  instrument,  —  a  man  nearly  a  century  in  advance  of 
his  day) ,  found  that  the  eclipses  of  the  satellites  showed  a  peculiar 
variation  in  their  times  of  occurrence,  which  he  explained  as  due  to 
the  time  taken  by  light  to  pass  through  space.  His  bold  and  original 
suggestion  was  rejected  by  most  astronomers  for  more  than  fifty 
years,  —  until  long  after  his  death,  —  when  Bradley's  discovery  of 
aberration  (Art.  225)  proved  the  correctness  of  his  views. 

629.  If  the  planet  and  earth  remained  at  an  invariable  distance 
the  eclipses  of  the  satellites  would  recur  with  unvarying  regularity 
(their  disturbances  being  very  slight) ,  and  the  mean  interval  could 
be  determined,  and  the  times  tabulated.  But  if  we  thus  predict  the 
times  of  eclipses  for  a  synodic  period  of  the  planet,  then,  begin- 
ning at  the  time  of  opposition, 
it  will  be  found  that  as  the 
planet  recedes  from  the  earth, 
the  eclipses  fall  constantly  more 
and  more  behindhand,  and  by 
precisely  the  same  amount  for 
all  four  of  the  satellites.  The 
difference  between  the  tabulated 
and  observed  time  continues  to 
increase  until  the  planet  is  near 
conjunction,  when  the  eclipses 
are  more  than  sixteen  minutes 
late. 


From  the  insufficient  observa- 
tions at  his  command,  Roemer 
made  the  difference  twenty-two 
minutes. 


Fia.  181. 
Determination  of  the  Equation  of  Light. 


After  the  conjunction,  the  eclipses  quicken  their  pace  and  exactly 
make  up  all  the  loss ;  so  that  when  opposition  is  reached  once  more, 
they  are  again  on  time. 

It  is  easy  to  see  from  Fig.  181  that  at  opposition  the  planet  is 
nearer  the  earth  than  at  conjunction  by  just  twice  the  radius  of  the 
earth's  orbit ;  i.e. ,  JB  —  JA  =  2  SA.  The  whole  apparent  retardation 
of  the  eclipses  between  opposition  and  conjunction,  should  therefore 
be  exactly  twice  the  time  required  for  light  to  come  from  the  sun  to 
the  earth.  This  time  is  very  nearly  500  seconds,  or  8m  20*. 


388  JUPITER. 

Early  in  the  century  Delambre,  from  all  the  satellite  eclipses  of  which  he 
could  then  secure  observations,  found  it  to  be  493s.  A  few  years  ago  a 
redetermination  by  Glasenapp  of  Pulkowa  made  it  501",  from  fifteen  years' 
observation  of  the  eclipses  of  the  first  satellite.  The  value  at  present 
accepted  is  499",  and  can  hardly  be  erroneous  by  more  than  1s. 

630.  Photometric  Observations  of  the  Eclipses.  —  The  eclipses  are 
gradual  phenomena,  the  obscuration  of  the  satellite  proceeding  con- 
tinuously from  the  time  it  first  strikes  the  shadow  of  the  planet  until 
it  entirely  vanishes.     The  moment  at  which  the  satellite  seems  to 
disappear  depends,   therefore,   on  the  state  of  the   air  and  of  the 
observer's  eye,  and  upon  the  power  of  his  telescope.     The  same  is 
true  of  the  reappearance ;  so  that  the  observations  are  doubtful  to 
the  extent  of  from  half  a  minute  for  the  first  satellite  (which  moves 
quickly),  to  a  full  minute  for  the  fourth.     Professor  Pickering  has 
proposed  to  substitute  for  this  comparatively  indefinite   moment  of 
disappearance  or  reappearance,  the  instant  when  the  satellite  has  lost 
or  regained  just  half  its  normal  light,  and  he  determines  this  instant 
by  a  series  of  photometric  comparisons  with  one  of  the  neighboring 
uneclipsed  satellites,  or  with  the  planet  itself. 

These  comparisons  are  made  with  a  special  photometer  devised  for  the 
purpose,  and  planned  with  reference  to  rapid  reading  :  by  merely  turning  a 
small  button,  the  observer  is  immediately  able  to  make  the  image  of  the 
uneclipsed  satellite  appear  to  be  of  the  same  brightness  as  the  satellite  which 
is  disappearing,  and  the  observations  can  be  repeated  very  rapidly  with  the 
help  of  special  contrivances  for  recording  the  times  and  readings.  It  is 
found  that  this  instant  of  "  half-brightness  "  can  be  deduced  from  the  set  of 
photometric  readings  with  an  error  not  much  exceeding  a  second  or  two. 
Observations  of  this  kind  have  now  been  going  at  Cambridge  (U.  S.)  1  for 
several  years.  A  similar  plan  has  also  been  devised  by  Cornu,  and  is  being 
carried  out  at  the  Paris  Observatory  under  his  direction. 

A  series  of  such  observations  covering  the  planet's  whole  period  of  twelve 
years,  ought  to  give  us  a  much  more  accurate  determination  of  the  light- 
equation  than  we  now  have. 

631.  Until  1849  our  only  knowledge  of  the  velocity  of  light  was 
obtained  by  observations  of  Jupiter's   satellites.     By  assuming   as 

1  Professor  Pickering  has  more  recently  applied  a  photographic  process  to 
these  observations  with  most  gratifying  success.  A  series  of  pictures  is  taken, 
each  with  an  exposure  of  10s,  the  time  being  recorded  on  a  chronograph,  and 
they  determine  with  great  precision  the  moment  when  the  satellite's  brightness 
had  any  special  value,  say  fifty  per  cent  of  its  maximum. 


SATURN.  389 

known  the  earth's  distance  from  the  sun,  the  velocity  of  light 
follows  when  we  know  the  time  occupied  by  light  in  coming  from 
the  sun.  At  present,  however,  the  case  is  reversed :  we  can  deter- 
mine the  velocity  of  light  by  two  independent  experimental  methods, 
and  with  a  surprising  degree  of  accuracy ;  and  then,  knowing  the 
velocity  and  the  light-equation,  we  can  deduce  the  distance  of  the  sun. 

SATURN. 

632.  The  Orbit  and  Period.  —  Saturn   is   the  remotest  of  the 
ancient  planets,  its  mean  distance  from  the  sun  being  9.54  astro- 
nomical units,  or  886000000  miles.      The   actual   distance  varies, 
however,  by  nearly  50  000000  miles  on  account  of  the  eccentricity  of 
its  orbit  (0.056),  which  is  a  little  greater  than  that  of  Jupiter. 

Its  nearest  approach  to  the  earth  at  a  December  opposition  (the 
longitude  of  its  perihelion  being  90°  4')  is  744  millions  of  miles,  and 
its  greatest  distance  at  a  May  conjunction  is  1028  millions.  It 
is  so  far  from  the  sun  that  these  changes  of  distance  do  not  so 
greatly  affect  its  apparent  brightness,  as  in  the  case  of  the  nearer 
planets,  the  whole  range  of  variation  from  this  cause  being  less  than 
two  to  one  ;  that  is,  at  the  nearest  of  all  oppositions,  the  planet  is 
not  twice  as  bright  as  the  remotest  of  all  conjunctions.  The 
changing  phases  of  the  rings  make  quite  as  great  a  difference  as  the 
variations  of  distance. 

The  orbit  is  inclined  to  the  ecliptic  about  2-J-0. 

The  sidereal  period  of  the  planet  is  twenty-nine  and  one-half  years, 
the  synodic  period  being  378  days. 

The  planet  itself  is  unique  among  the  heavenly  bodies.  The  great 
belted  globe  carries  with  it  a  retinue  of  ten  satellites,  and  is  sur- 
rounded by  a  system  of  rings  unlike  anything  else  in  the  universe 
so  far  as  known,  the  whole  constituting  the  most  beautiful  and  most 
interesting  of  all  telescopic  objects. 

633.  Diameter,  Volume,  and  Surface. — The  apparent  mean  diam- 
eter of  the  planet  varies  from  20"  to  14"  according  to  the  distance. 
We  say  mean  diameter  because  this  planet  is  more  flattened  at  the 
pole  than  any  other,  its  ellipticity  being  nearly  ten  per  cent,  though 
different  observers  vary  somewhat  in  their  results.     The  equatorial 
diameter  of  the  planet  is  about  76470  miles,  and  its  polar  about 
69770,  the  mean  being  very  nearly  74200,  or  a  little  more  than  nine 
times  that  of  the  earth.     Its  surface  is  therefore  about  eighty-two 
times,  and  its  volume  820  times  that  of  the  earth. 


390  SATUKN. 

634.  Mass,  Density,  and  Gravity.  —  Its  mass  is  only  ninety-five 
times  the  earth's  mass,  from  which  follows  the  remarkable  fact  that 
the  density  of  Saturn  is  only  one-eighth  that  of  the  earth,  or  only 
about  five-sevenths  that  of  water.     It  is  by  far  the  least  dense  of  all 
the  planets.     The  superficial  gravity  is  1.2. 

635.  Axial  Rotation.  —  It  revolves  upon  its  axis  in  about  10h  14m 
according  to  a  determination  of  Professor  Hall,  made  in  1876  by 
means  of  a  white  spot  which  suddenly  appeared  upon  its  surface, 
and  continued  visible  for  some  weeks.     His  result  is  substantially 
confirmed  by  the  observations  of  Stanley  Williams  in  1893,  which, 
however,  appear  to  indicate  that  spots  in  different  latitudes  give 
rotation-periods  which  differ  slightly,  but  systematically. 

The  inclination  of  the  axis  to  the  planet's  orbit  is  about  27°. 

636.  Surface,  Albedo,  and  Spectrum.  —  As  in  the  case  of  Jupiter, 
the  edges  of  the  disc  are  not  quite  so  brilliant  as  the  central  por- 
tions, so  that  the  belts  appear  to  fade  out  near  the  limb.     These 
belts  are  less  distinct  and  less  variable  than  those  of  Jupiter ;  and 
are  arranged  as  shown  in  Fig.  182,  with  a  very  brilliant  zone  at 
the  equator,  though  the  engraving  much  exaggerates  the  contrast. 
The  planet's  pole  is  sometimes  marked  by  a  darkish  cap  of  greenish 
hue. 

According  to  Zollner,  the  Albedo,  or  reflecting  power  of  the  surface, 
is  0.52,  almost  precisely  the  same  as  that  of  Venus,  but  a  little  infe- 
rior to  that  of  Jupiter.  The  spectrum  of  the  planet  is  the  solar  spec- 
trum without  any  evidence  of  the  presence  of  water-vapor,  so  far  as 
can  be  made  out,  but  with  certain  unexplained  dark  bands  in  the  red 
and  orange  similar  to  those  observed  in  the  spectrum  of  Jupiter. 
The  darkest  of  these  bands,  however,  are  not  seen  in  the  spectrum 
of  the  ring ;  this  might  have  been  expected,  since  the  ring  probably 
has  but  little  atmosphere. 

637.  The  Rings.  —  The  most  remarkable  peculiarity  of  Saturn  is 
his  ring-system.     The  planet  is  surrounded  by  three,  thin,  flat,  con- 
centric rings  like  circular  discs  of  paper  pierced  through  the  centre. 
Two  of  them  are  bright,  while  the  third,  the  one  nearest  to  the 
planet,  is  dusky  and  comparatively  difficult  to  see.     They  are  gener- 
ally referred  to  by  Struve's  notation  as  A,  B,  and  C,  A  being  the 
exterior  one. 

For  nearly  fifty  years  this  appendage  of  Saturn  was  a  complete 
enigma  to  astronomers.  Galileo,  in  1610,  saw  with  his  little  tele- 


SATURN'S  RINGS. 


391 


scope  that  the  planet  appeared  to  have  something  attached  to  it  on 
each  side,  and  he  announced  the  discovery  that  "the  outermost 
planet  is  triple/'  —  "  ultimam  planetam  tergeminam  observavi." 


FIG.  182.  — Saturn  and  his  Rings. 

Not  long  afterwards  the  rings  were  edgewise  to  the  earth  so  that  they 
became  invisible  to  him  ;  and  in  his  perplexity  he  inquired  "  whether 
Saturn  had  devoured  his  children,  according  to  the  legend."  Huy- 


392  SATURN. 

ghens,  in  1655,  was  the  first  to  solve  the  problem  and  explain  the  true 
structure  of  the  rings.  Cassini,  twenty  years  later,  discovered  that 
the  ring  was  double,  —  composed  of  two  concentric  portions  with  a 
narrow  black  rift  of  division  between  them. 

The  third,  or  dusky  ring,  (7,  is  an  American  discovery,  and  was 
first  brought  to  light  by  W.  C.  Bond  at  Cambridge,  U.  S.,  in  Novem- 
ber, 1850.  About  two  weeks  later,  but  before  the  news  had  been 
published  in  England,  it  was  also  discovered  independently  by 
Dawes. 

For  a  while  there  was  some  question  whether  it  was  not  really  a  new 
formation ;  but  an  examination  of  old  drawings  shows  that  Herschel  and 
several  other  astronomers  had  previously  seen  it  where  it  crosses  the  planet, 
although  without  recognizing  its  character. 

638.  Dimensions  of  the  Rings — The  outer  ring,  A,  has  an  exterior 
diameter  of  172000  miles  (Barnard),  and  is  nearly  11000  miles  wide.     The 
division  between  it  and  ring  B  is  about  2300  miles  in  width,  and  apparently 
perfectly  uniform  all  around.     Ring  B  is  about  18000  miles  wide,  and  is 
much  brighter  than  A,  especially  at  its  outer  edge.     At  the  inner  edge  it 
becomes  less  brilliant,  and  is  joined  without  any  sharp  line  of  demarcation 
by  ring  C,  which  is  sometimes  known  as  the  "  gauze  "  or  "  crape  "  ring, 
because  it  is  only  feebly  luminous  and  is  semi-transparent,  allowing  the 
edge  of  the  planet  to  be  seen  through  it.     The  innermost  ring  is  nearly,  per- 
haps not  quite,  as  wide  as  the  outer  one,  A.     There  is  thus  left  a  clear  space 
of  from  5000  to  6000  miles  in  width  between  the  planet's  equator  and  the 
inner  edge  of  the  gauze  ring,  the  whole  ring  system  having  an  external 
diameter  of  172000  miles,  and  a  width  of  about  42000. 

The  thickness  of  the  rings  is  very  small  indeed,  probably  not  ex- 
ceeding 100  miles.  If  we  were  to  construct  a  model  of  them  on  the 
scale  of  10,000  miles  to  the  inch,  so  that  the  outer  one  would  be  nearly 
seventeen  inches  in  diameter,  the  thickness  of  an  ordinary  sheet  of 
writing  paper  would  be  about  in  due  proportion.  This  extreme  thin- 
ness is  proved  by  the  appearances  presented  when  the  plane  of  the 
ring  is  directed  towards  the  earth,  as  it  is  once  in  every  fifteen  years. 
At  that  time  the  ring  becomes  invisible  for  several  days  even  to 
the  most  powerful  telescopes. 

639,  Phases  of  the  Rings.  —  The  rings  are  parallel  to  the  equator 
of  the  planet,  which  is  inclined  about  27°  to  its  orbit,  and  about  28° 
to  the  plane  of  the  ecliptic,  the  two  nodes  of  the  ring  being  in  longi- 
tude 168°  and  348°,  in  the  constellations  of  Aquarius  and  Leo.    Now 


PHASES   OF   THE   KINGS.  393 

in  the  planet's  revolution  around  the  sun,  the  plane  of  the  planet's 
equator  and  of  the  rings  always  keeps  parallel  to  itself  (as  shown 
in  Fig.  183),  just  as  does  the  plane  of  the  earth's  equator.  Twice, 
therefore,  in  the  planet's  revolution,  when  the  plane  of  the  ring 


FIG.  183.— The  Phases  of  Saturn's  Rings. 

passes  through  the  earth,  we  see  it  edgewise;1  and  twice  at  its  maxi- 
mum width,  when  it  is  at  the  points  half-way  between  the  nodes.  The 
angle  of  inclination  being  28°,  the  apparent  width  of  the  ring  at  the 
maximum  is  just  about  half  its  length.  The  last  disappearance  of 
the  rings  was  in  October,  1907 ;  the  next  will  be  in  the  summer  of 
1922.  Near  the  time  of  disappearance  the  ring  appears  simply  as 
a  thin  needle  of  light  projecting  on  each  side  of  the  planet  to  a 
distance  nearly  equal  to  its  diameter.  Upon  this  the  satellites  are 
threaded  like  beads  when  they  pass  behind  or  in  front  of  it. 

640.  Irregularities  of  Surface  and  Structure. — When  the  rings 
are  edgewise  we  find  that  there  are  notable  irregularities  upon  them, 
such  as  the  condensations  or  "  knots  "  observed  at  the  1907  disap- 
pearance. 

The  same  thing  is  indicated  by  certain  peculiarities  sometimes  reported 
in  the  form  of  the  shadow  cast  by  the  planet  on  the  rings ;  but  caution  must 
be  used  in  accepting  and  interpreting  such  observations,  because  illusions 

1  In  traversing  the  earth's  of  bit  the  plane  of  the  ring  occupies  about  359.6 
days,  during  which  the  earth  crosses  it  either  once  or  three  times,  according  to  cir- 
cumstances. The  ring  then  becomes  absolutely  invisible  to  all  existing  telescopes 
for  several  days ;  nor  can  it  be  seen  by  any  but  very  powerful  instruments  during 
the  time  while  the  plane  lies  between  the  earth  and  sun,  often  for  several  weeks. 
There  are  usually  two  such  "periods  of  disappearance"  during  the  critical  year. 


394  SATURN. 

are  very  apt  to  occur  from  the  least  indistinctness  of  vision  or  feebleness  of 
light.  The  writer  has  usually  found  that  the  better  the  seeing,  the  fewer 
abnormal  appearances  were  noted,  and  the  experience  of  the  Washington 
observers  is  the  same. 

It  can  hardly  be  doubted  that  the  details  of  the  rings  are  continu- 
ally changing  to  some  extent.  Thus  the  outer  ring,  A,  is  occasion- 
ally divided  into  two  by  a  very  narrow  black  line  known  as  "  Encke's 
division,"  although  more  usually  there  is  merely  a  darkish  streak 
upon  it,  not  amounting  to  a  real  ' '  crack  "  in  the  surface. 

641.  Structure  of  the  Rings.  —  It  is  now  universally  admitted 
that  the  rings  are  not  continuous  sheets  of  either  solid  or  liquid 
matter,  but  are  composed  of  a  swarm  of  separate  particles,  each  a 
little  independent  moon  pursuing  its  own  path  around  the  planet. 
The  idea  was  suggested  long  ago,  by  J.  Cassini  in  1715,  and  by 
Wright  in  1750,  but  was  lost  sight  of  until  Bond  revived  it  in  con- 
nection with  his  discovery  of  the  dusky  ring.  Professor  Benjamin 
Peirce  soon  afterwards  demonstrated  that  the  rings  could  not  be  con- 
tinuous solids  ;  and  Clerk  Maxwell  finally  showed  that  they  can  be 
neither  solid  nor  liquid  sheets,  but  that  all  the  known  conditions  would 
be  answered  by  supposing  them  to  consist  of  a  flock  of  separate  and 
independent  bodies,  moving  in  orbits  nearly  circular  and  in  one 
plane,  —  in  fact,  a  swarm  of  meteors. 

641*.  This  «  Meteoric  Theory  "  has  recently  (in  1895)  been  beautifully  con- 
firmed by  the  spectroscopic  observations  of  Keeler,  illustrated  in  Fig.  183  *. 
Photographs  were  made  of  the  spectrum  of  the  planet  and  its  rings  with  the 
slit  of  the  spectroscope  crossing  the  planet's  image,  as  shown  in  the  figure. 
At  the  western  limb  of  the  planet  and  extremity  of  the  ring  the  motion  of 
rotation  was  carrying  the  particles  from  us,  and  the  displacement  of  the 
spectrum  lines  should  be  towards  the  red,  according  to  Doppler's  principle 
(we  note  also  in  passing  that,  since  the  particles  shine  by  reflected  sunlight, 
the  displacement  is  practically  doubled,  being  twice  as  great  as  if  they  were 
self-luminous).  On  the  eastern  side  the  shift  is  towards  the  violet.  Now, 
on  looking  at  the  diagram  of  the  spectrum,  given  below  the  planet,  we  see 
that,  while  at  C  the  line  in  the  spectrum  is  displaced  redwards,  as  it  ought 
to  be,  the  displacement  at  the  outer  edge  of  the  ring  is  less  than  at  the  inner;  and 
correspondingly  at  A.  This  shows  that,  as  theory  requires,  the  outer  edge 
revolves  more  slowly  than  the  inner.  The  fact  is  made  conspicuous  by  its  effect 
upon  the  inclination  of  the  lines :  while  in  the  spectrum  of  the  ball  the  lines 
slope  upwards  towards  the  right,  in  the  ring-spectrum  they  slope  the  other 
way.  The  observation  is  very  delicate,  as  the  whole  width  of  the  spectrum 


THEORY    OF    THE    RINGS. 


395 


1  Millimetre 


was  not  quite  a  millimetre  (the  figure  being  magnified  nearly  fifty  times) ; 
but    Keeler's    results    have 
since   been   fully  confirmed 
by  Deslandres,    Belopolsky, 
and  Campbell. 

An  independent  photo- 
metric confirmation  has  been 
derived  by  Seeliger  from  the 
way  in  which  the  apparent 
brightness  of  the  rings  varies 
with  their  phases  ;  and  an- 
other from  the  behavior  of 
lapetus  (the  outer  satellite) ? 
as  observed  by  Barnard  in 
1892  while  undergoing 
eclipse.  The  satellite  van- 
ished completely  in  passing 
through  the  shadow  of  the 
ball  and  bright  rings,  but  re- 
appeared when  immersed  in 
that  of  the  semi-transparent 
dusky  ring. 

The  investigations  of  Hermann  Struve  show  that  the  mass  of  the  rings  is 
inappreciable  :  they  produce  no  observable  effect  upon  the  motion  of  the 
satellites.  To  use  his  graphic  expression,  "  they  seem  to  be  composed  of 
immaterial  light,"  —  mere  dust-films  or  wreaths  of  fog. 


50  &m 


50fcm. 


FIG.  183  *.  —  Spectroscopic  Observation  of 
Saturn's  Ring  (Keeler). 


642.  Stability  of  the  Ring".  —  If  the  ring  were  solid  it  would  cer- 
tainly not  be  stable,  and  the  least  disturbance  would  bring  it  down  upon 
the  planet ;  nor  is  it  certain  that  even  the  swarm-like  structure  makes  it 
forever  secure.  It  is  impossible  to  say  positively  that  the  rings  may  not 
after  a  time  be  broken  up.  A  few  years  ago  there  was  much  interest  in  a 
speculation  which  Struve  published  in  1851.  All  the  measures  which  he 
could  obtain  up  to  that  date  appeared  to  show  that  a  change  was  actually 
in  progress,  and  that  the  inner  edge  of  the  ring  was  extending  itself  towards 
the  planet.  His  latest  series  of  measurements  (in  1885)  does  not,  however, 
confirm  this  theory.  They  show  no  considerable  change  since  1850,  and  the 
measurements  of  other  observers  agree  with  his  in  this  respect. 

The  researches  of  Professor  Kirkwood  of  Indiana  make  it  probable  that 
the  divisions  in  the  ring  are  due  to  the  perturbations  produced  by  the  satel- 
lites. They  occur  at  distances  from  the  planet  where  the  period  of  a  small 
body  would  be  precisely  commensurable  with  the  periods  of  a  number  of 
the  satellites.  It  will  be  remembered  that  similar  gaps  are  found  in  the 
distribution  of  the  asteroids,  at  points  where  the  period  of  an  asteroid  would 
be  commensurable  with  that  of  Jupiter. 


396  SATURN'S  SATELLITES. 

643.  Satellites.  —  Saturn  has  ten1  of  these  attendants.  The 
largest  of  them  was  discovered  by  Huyghens  in  1655.  It  appears  as 
a  star  of  the  ninth  magnitude,  and  is  easily  observable  with  a  three- 
inch  telescope.  Four  others  were  discovered  by  Cassini  before 
1700,  two  by  Sir  William  Herschel  near  the  end  of  the  last  century, 
and  one,  Hyperion,  by  W.  C.  Bond  of  Cambridge,  in  September, 
1848,  and  independently  by  Lassell  at  Liverpool  two  days  later. 
(For  Phoebe  and  Themis,  the  two  newest,  see  note  on  page  406.) 

The  range  of  the  system  is  enormous.  lapetus  has  a  distance  of 
2,225000  miles,  with  a  period  of  79  days,  nearly  as  long  as  that  of 
Mercury.  There  is  a  remarkable  variation  in  the  brightness  of  this 
satellite,  y  On  the  western  side  of  the  planet  it  is  fully  twice  as  bright 
as  upon  the  eastern,  which  practically  demonstrates  that,  like  our  own 
moon,  it  keeps  the  same  face  towards  the  planet  at  all  times,  one-half 
of  its  surface  being  much  more  brilliant  than  the  other. 

Mimas,  the  nearest  and  smallest  of  the  satellites,  coasts  around  the 
edge  of  the  ring  at  a  distance  from  it  of  only  34,000  miles,  or 
118,000  from  the  planet's  centre,  having  a  period  of  only  22-J-  hours. 
This  satellite  is  so  small  and  so  near  the  planet  that  it  can  be  seen 
only  by  very  large  telescopes  and  under  favorable  conditions. 

Titan,  as  its  name  suggests,  is  by  far  the  largest  of  the  family. 
Its  distance  is  about  770,000  miles,  and  its  period  a  little  less  than 
16  days.  Its  diameter,  as  measured  by  Barnard,  is  2720  miles,  and 
according  to  Stone,  its  mass  is  ^Vu  of  Saturn's. 


644.  Peculiar  Behavior  of  Hyperion.  —  Hyperion  has  a  distance 
of  934,000  miles,  and  a  period  of  21|  days.  Under  the  action  of  Titan  its 
orbit  is  rendered  considerably  eccentric,  and  its  line  of  apsides  always  keeps 
itself  in  the  line  of  conjunction  with  Titan,  retrograding  in  a  way  which 
at  first  seemed  to  defy  theoretical  explanation,  but  turns  out  to  be  only 
a  "  new  case  in  celestial  mechanics,"  and  a  necessary  result  of  the  disturb- 
ance by  Titan.  Mimas  also  undergoes  a  very  considerable  disturbance,  which 
alternately  accelerates  and  retards  it  to  the  extent  of  nearly  60°  of  its  orbit. 

1  Until  Herschel's  time  it  was  customary  to  distinguish  the  satellites  as  first, 
second,  etc.,  in  order  of  distance  from  the  planet  ;  but  as  Herschel's  new  satellites 
were  within  the  orbits  of  those  which  were  known  before,  their  discovery  con- 
fused matters,  and  the  confusion  became  worse  confounded  when  the  eighth 
appeared.  They  are  now  usually  designated  by  names  assigned  by  Sir  John 
Herschel  as  follows,  beginning  with  the  most  remote,  namely  :  lapgtus  (Hype- 
rion), Titan;  Rhea,  Dione,  Tethys;  Enceladus,  Mimas.  It  will  be  noticed  that 
these  names,  leaving  out  Hyperion,  which  was  undiscovered  when  they  were 
assigned,  form  a  line  and  a  half  of  a  regular  Latin  pentameter. 


URANUS.  397 

The  orbit  of  lapetus  is  inclined  about  10°  to  the  plane  of  the  rings,  but 
all  the  other  satellites1  move  exactly  in  their  plane,  and  all  the  five  inner 
ones  move  in.  orbits  sensibly  circular.  The  orbits  of  lapetus,  Hyperion,  and 
Titan  have  a  slight  eccentricity. 

URANUS. 

645.  As  the  satellites  of  Jupiter  were  the  first  heavenly  bodies 
to  be  "discovered/'  so  Uranus  was  the  first  "discovered"  planet, 
all  the  other  planets  tben  known  having  been  known  from  prehistoric 
antiquity.     On   March    13,  1781,  the   elder   Herschel,  in   sweeping 
over  the  heavens  systematically  with   a   seven-inch   reflector  made 
by  himself,  came  upon  an  object  which,  by  its  disc,  he  saw  at  once 
was  not  an  ordinary  star.     In  a  day  or  two  he  had  ascertained  that 
it  moved,  and  announced  the  discovery  as  that  of  a  comet.     After 
a  short  time,   however,  it  became  obvious  from  the  computations 
of  Lexell,   that  its  orbit  was  nearly  circular,  that   its  distance  was 
enormous,  and  that  its  path  did  not  at  all  resemble  that  ordinarily 
taken  by   a  comet ;  and  within  a  year  its   planetary  character  was 
recognized  and  it  was  formally  admitted  as  a  new  member  of  the 
solar  <system.     The  name  of  Uranus,  suggested  by  Bode,  finally  pre- 
vailed over  other  appellations  (Herschel  himself  called  it  the  Georgium 
Sidus,  in  honor  of  the  king) ,  with  the  symbol  $  or  §  .    The  former  is 
still  generally  used  by  English  astronomers. 

The  discovery  of  a  new  planet,  a  thing  then  utterly  unprecedented,  caused 
great  excitement.  The  king  knighted  Herschel,  gave  him  a  pension,  and 
furnished  him  with  the  funds  for  constructing  his  great  forty-foot  reflector 
of  four  feet  aperture,  with  which  he  afterwards  discovered  the  two  inner 
satellites  of  Saturn.  It  was  found  on  reckoning  back  from  the  date  of 
Herschel's  discovery  that  the  planet  had  been  several  times  before  observed 
as  a  star  by  astronomers  who  narrowly  missed  the  honor  which  fell  to  the 
more  fortunate  and  diligent  Herschel.  Twelve  such  observations  had  been 
made  by  Lemonnier  alone. 

646.  Orbit. — The  mean-distance  of  Uranus  from  the  sun  is  very 
nearly  1800  millions  of  miles,  and  the  eccentricity  a  trifle  less  than 
that  of  Jupiter's  orbit,  amounting  to  about  83,000000.     The  inclina- 
tion of  the  orbit  to  the  plane  of  the  ecliptic  is  very  slight,  only  46'. 
The  planet's  periodic  time  is  84  years,  and  the  synodic  period  (from 
opposition  to  opposition)  369d  16h.    The  orbital-velocity  is  4J  miles 
per  second. 

1  See  note  on  page  406. 


398  URANUS. 

647.  Appearance  and  Magnitude.  — Uranus  is  distinctly  visible  to 
the  naked  eye  on  a  dark  night  as  a  small  star  of  the  so-called  sixth 
magnitude.     It  is  so  remote,  its  orbit  having  a  diameter  more  than 
19  times  that  of  the  earth's,  that  there  is  very  little  change  in  its 
appearance,  and  it  makes  no  practical  difference  whether  it  is  at 
opposition  or  quadrature. 

In  the  telescope  it  shows  a  sea-green  disc  of  about  4"  in  apparent 
diameter,  corresponding  to  a  real  diameter  of  32,0001miles.  Its  sur- 
face is  about  16  times,  and  its  volume  about  66  times  greater  than 
that  of  the  earth,  so  that  the  earth  compares  in  size  with  Uranus 
about  as  the  moon  does  with  the  earth.  The  mass  of  Uranus  is  14.6 
times  that  of  the  earth,  and  its  density  and  surface-gravity  are  respec- 
tively 0.22  and  0.90. 

648.  Albedo  and  Light.  —  The  reflecting   power  of  the   planet's 
surface  is  very  high,  its  albedo,  according  to  Zollner,  being  0.64,  even 
exceeding  that  of  Jupiter.     It  is  to  be  remembered,  however,  that 
sunlight  at  Uranus  is  only  ^|¥  as  intense  as  at  the  earth,  and  only 
about  y1^  as  intense  as  at  Jupiter ;  so  that  the  disc  of  the  planet  does 
not  appear  in  the  telescope  even  nearly  as  bright  as  a  piece  of  Jupiter's 
disc  of  the  same  apparent  size.     The  greenish  blue  tint  of  the  planet 
is  accounted  for  by  the  fact  that  its  spectrum  shows  certain  conspicu- 
ous dark  bands  in  its  lower  portion,  bands    perhaps  identical  with 
those  which  are  visible  in  the  spectrum  of  Saturn,  but  much  more 
intense.     These  facts  probably  indicate  a  dense  atmosphere. 

649.  Polar  Compression,  Belts,  and  Rotation.  —  The  disc  of  the 
planet  shows  a  decided  ellipticity — about  Tx¥  according  to  the  Prince- 
ton observations  of  1883,  which  agree  nearly  with  those  of  Schiapa- 
relli,  and  have  since  been  confirmed  by  Barnard  at  the  Lick  Ob- 
servatory.    There  are  also  sometimes  visible  upon  the  planet's  disc 
certain  extremely  faint  bands  or  belts,  much  like  the  belts  of  Jupiter 
viewed  with  a  very  small  telescope.     What  is  exceedingly  singular, 
however,  is  that  the  trend  of  these  belts  seems  to  indicate  a  plane 
of  rotation  not  coinciding  with   the  plane   of   the   satellites9   orbits. 
Nearly  all  the  observers  who  have  seen  them  at  all  find  that  they  are 
inclined  to  the  satellites'  orbit-plane  at  an  angle  of  from  15°  to  40°. 
Now  unless  there  is  some  error  in  the  investigations  of  La  Place 
upon  the  motions  of  satellites,  it  is  probable  that  the  plane  of  these 
orbits  does  very  nearly  coincide  with  that  of  the  planet's  equator. 
Probably  the  error  lies  in  judging  the  direction  of  the  belts,  which 
at  the  best  are  at  the  very  limit  of  visibility. 

1  See  second  note  on  page  406. 


SATELLITES    OF    URANUS.  399 

One  or  two  observers  have  assigned  to  the  planet  rotation  periods 
ranging  from  9h  to  12h  ;  but  it  cannot  be  said  that  any  determination 
of  this  element  yet  made  is  to  be  trusted. 

650.  Satellites.  —  The   planet   has   four   satellites;   viz.,   Ariel, 
Umbriel,  Titania,  and  Oberon  ;  Ariel  being  the  nearest  to  the  planet. 
The  two  brightest  of  them,  Oberon  and  Titania,  were  discovered  by 
Sir  William  Herschel  a  few  years  after  the  discovery  of  the  planet. 
He  observed  them  sufficiently  to  obtain  a  reasonably  correct  determi- 
nation of  their  distances  and  periods. 

It  is  not  certain  that  he  saw  either  of  the  other  two,  though  he  thought  lie 
had  found  six  satellites  in  all,  and  a  few  years  ago  a  popular  writer  on 
astronomy  actually  credited  the  planet  with  eight  satellites,  —  the  four 
whose  names  have  been  given,  and  four  others  which  Herschel  supposed  he 
had  seen. 

Ariel  and  Umbriel  were  first  certainly  discovered  by  Lassell  in  1851,  and 
have  since  been  satisfactorily  observed  by  numerous  large  telescopes.  They 
are  telescopically  the  smallest  bodies  in  the  solar  system,  and  the  most 
difficult  to  see.  In  real  size,  they  are,  of  course,  much  larger  than  the  satel- 
lites of  Mars  or  many  of  the  asteroids,  very  likely  measuring  from  200  to 
500  miles  in  diameter ;  but  they  are  ten  times  as  far  away  as  the  asteroids, 
and  illuminated  by  a  sunlight  not  ^  as  brilliant  as  theirs. 

651.  Satellite  Orbits.  — The  orbits  of  the  satellites  are  sensibly  circu- 
lar, and  all  lie  in  one  plane,  which,  as  has  been  said,  ought  to  be,  and  prob- 
ably is,  coincident  with  the  plane  of  the  planet's  equator.     They  are  very 
close-packed  also,  Oberon  having  a  distance  of  only  375,000  miles,  with  a 
period  of  13d  llh,  while  Ariel  has  a  period  of  2d  12h,  at  a  distance  of  120,000 
miles.     Titania,  the  largest  and  brightest  of  them,  has  a  distance  of  280,000 
miles,  somewhat  greater  than  that  of  the  moon  from  the  earth,  with  a  period 
of  8d  17h.    Under  favorable  circumstances  this  satellite  can  be  just  seen  with 
a  telescope  of  eight  or  nine  inches  aperture. 

652.  Plane   of  Revolution.  —  The  most  remarkable  thing   about 
this  satellite  system  remains  to  be  mentioned.     The  plane  of  their 
orbits  is  inclined  82°. 2  to  the  plane  of  the  ecliptic,  and  in  that  plane 
they  revolve  backwards;   or  we  may  say,  what  comes  to  the  same 
thing,  that  their  orbits  are  inclined  to  the  ecliptic  at  an  angle  of 
97°. 8,  in  which  case  their  revolution  is  to  be  considered  as  direct. 

When  the  line  of  nodes  of  their  orbit  plane  passes  through  the  earth, 
as  it  did  in  1840  and  1882,  the  orbits  are  seen  edgewise  and  appear  as 
straight  lines.  On  the  other  hand,  in  1861,  they  were  seen  almost  in  plan 


400  NEPTUNE. 

as  nearly  perfect  circles,  and  were  seen  so  again  in  1903.  The  year 
1882-83  was  a  specially  favorable  time  for  determining  the  inclination  of 
the  orbits  and  the  position  of  the  nodes,  as  well  as  for  measuring  the  polar 
compression  of  the  planet. 

NEPTUNE. 

653,  The  discovery  of   this  planet  is  justly  reckoned  as  the 
greatest  triumph  of  mathematical  astronomy.    Uranus  failed  to  move 
precisely  in  the  path  which  the  computers  predicted  for  it,  and  was 
misguided  by  some  unknown  influence  to  an  extent  which  a  keen  eye 
might  almost  see  without  telescopic  aid.     The  difference  between  its 
observed  place  and  that  prescribed  for  it  had  become  in  1845  nearly 
as  much  as  the  "  intolerable  "  quantity  of  2'  of  arc. 

The  following  illustration  will  show  how  extremely  small  was  this  dis. 
crepancy  which  the  astronomers  considered  to  be  "  intolerable." 

Near  the  bright  star  Vega  there  are  two  little  stars  which  form  with  it  a 
small  equilateral  triangle,  the  sides  of  the  triangle  being  about  1|°  long. 
The  northern  one  of  the  two  little  stars  is  the  beautiful  double-double  star 
c  Lyrae,  and  can  be  seen  as  double  by  a  keen  eye  without  a  telescope,  the 
two  companions  being  about  3£'  apart.  Now  the  distance  between  the  com- 
puted place  of  Uranus  and  its  actual  position  was,  when  at  its  maximum, 
just  a  little  more  than  half  of  the  distance  between  these  components  of  e 
Lyrse,  that  only  a  keen  eye  can  separate.  One  would  almost  say  that  such 
a  difference  was  hardly  worth  minding. 

But  just  these  minute  discrepancies  constituted  the  data  which 
were  found  sufficient  for  calculating  the  position  of  a  hitherto 
unknown  planet,  and  bringing  it  to  light.  Leverrier  wrote  to  Galle, 
in  substance:  "Direct  your  telescope  to  a  point  on  the  ecliptic  in  the 
constellation  of  Aquarius,  in  longitude  326°,  and  you  will  find  within 
a  degree  of  that  place  a  new  planet,  looking  like  a  star  of  about  the 
ninth  magnitude,  and  having  a  perceptible  disc."  The  planet  was 
found  at  Berlin  "on  the  night  of  Sept.  23,  1846,  in  exact  accordance 
with  this  prediction,  within  half  an  hour  after  the  astronomers  began 
looking  for  it,  and  only  about  52'  distant  from  the  precise  point  that 
Leverrier  had  indicated. 

654.  So  far  as  the  mathematical  operations  are  concerned,  the 
honor  is  to  be  equally  divided  between  two  then  young  men,  — 
Leverrier  of  Paris,  and  Adams  of  Cambridge,  England.     Each  took 
up  the  problem,  and  by  perfectly  independent  and  considerably  dif- 
ferent methods  arrived  at  substantially  the  same  solution,  and  each 


DISCOVERY    OF    NEPTUNE.  401 

promptly  communicated  the  result  (Adams  some  months  earlier  than 
Leverrier)  to  a  practical  astronomer  provided  with  the  necessary 
apparatus  for  actually  detecting  the  planet. 

Adams,  who  was  then  a  graduate  of  three  years'  standing,  a  fellow  and  a 
tutor  in  his  college,  communicated  his  results  to  Challis,  his  professor  of 
astronomy  at  Cambridge,  in  the  autumn  of  1845.  Challis  at  once  con- 
sulted Airy,  the  Astronomer  Royal,  but  between  them  the  matter  rather  lay 
in  abeyance  for  some  months,  until  a  notice  appeared  of  a  preliminary  paper 
by  Leverrier,  which  indicated  that  he  also  had  reached  substantially  the 
same  conclusions  as  Adams.  Then,  at  the  urgent  suggestion  of  Airy, 
Challis  decided  to  begin  the  search  at  once,  and  to  capture  the  planet  by 
siege,  so  to  speak.  If  he  had  had  such  star-maps  as  we  now  possess  of  the 
regions  where  the  planet  lay  concealed,  it  would  have  been  comparatively  an 
easy  operation ;  but  as  he  had  not,  he  decided  to  go  over  a  space  10°  wide 
by  30°  long,  and  to  go  over  it  three  times.  The  positions  of  all  fixed 
stars  would  of  course  be  the  same  at  each  of  the  three  observations,  but  a 
planet  would  change  its  place  in  the  meantime,  and  so  would  be  surely 
detected. 

He  began  his  work  on  July  29,  including  in  his  sweep  all  stars  down  to 
the  tenth  magnitude.  When,  on  Oct.  1,  he  learned  of  the  actual  discovery 
of  the  planet,  he  had  recorded  the  positions  of  something  over  3000  stars, 
and  was  preparing  to  map  them.  He  had  already  secured,  as  it  turned  out, 
three  observations  of  the  planet  on  Aug.  4,  Aug.  12,  and  Sept.  29,  and 
of  course  it  was  only  the  question  of  a  few  weeks  more  or  less  when  the 
discussion  of  the  observations  would  have  brought  the  planet  to  light. 

But  while  this  rather  deliberate  process  was  going  on  in  England,  Lever- 
rier had  revised  his  work,  making  a  second  approximation,  and  had  commu- 
nicated his  results  to  Galle,  at  Berlin,  substantially  as  above  indicated.  The 
Berlin  astronomers  had  the  great  advantage  of  a  new  star-chart  by  Bremiker, 
covering  that  very  region  of  the  sky,  and  therefore  did  not  need  to  enter 
upon  any  such  tedious  campaign  as  that  begun  by  Challis.  In  less  than 
half  an  hour  they  found  a  new  star,  not  indicated  on  the  map,  and  showing 
a  sensible  disc,  just  as  Leverrier  had  predicted ;  and  within  twenty-four  hours 
its  motion  proved  it  to  be  the  planet. 

655 .  Computed  Elements  Erroneous.  —  Both  Adams  and  Leverrier, 
besides  computing  the  planet's  position  in  the  sky  had  deduced 
elements  of  its  orbit,  and  a  value  for  its  mass,  which  turned  out  to  be 
considerably  erroneous.  The  reason  was  that  they  had  assumed  that 
the  mean  distance  of  the  new  planet  from  the  sun  would  follow  Bode's 
law,  a  supposition  which,  as  it  turned  out,  is  not  even  roughly  true, 
although  it  was  entirely  warranted  by  the  existing  facts,  since  all  the 
then  known  planets,  not  excepting  Uranus,  obey  it  with  reasonable 


402  %     NEPTUNE. 

exactness.  This  assumption  of  an  erroneous  mean  distance  of  thirty- 
eight  astronomical  units,  instead  of  the  true  distance  of  thirty,  carried 
with  it  errors  in  all  the  other  elements  of  the  orbit ;  and  the  computed 
elements  are  so  wide  of  the  truth  that  great  authorities  have  main- 
tained that  the  actual  Neptune  was  not  at  all  the  Neptune  of  Leverrier 
and  Adams,  but  an  entirely  different  planet;  and  even  that  the 
discovery  was  a  "happy  accident."  It  was  not  an  accident  at  all, 
however.  While  the  data  and  methods  employed  were  not  competent 
to  determine  the  planet's  orbit  accurately,  they  were  sufficient  to 
determine  the  direction  of  the  unknown  body,  which  was  the  one 
thing  needed  to  insure  its  discovery.  The  computers  informed  the 
searchers  precisely  where  to  point  their  telescopes,  and  could  do  so 
again  were  a  new  case  of  the  same  kind  to  appear. 

656.  Old  Observations  of  Neptune.  —  After  a  few  weeks'  observa- 
tion of  the  new  planet  it  became  possible  to  compute  an  approximate  orbit ; 
and  reckoning  back  by  means  of  this  approximate  orbit,  the  approximate 
place  on  any  given  date  for  many  years  preceding  could   be   found.     On 
examining  the  observations  of  stars  made  by  different  astronomers  in  these 
regions  of  the  sky,  there  were  found  several  instances  in  which  they  had 
observed  the  planet ;  a  star  of  the  ninth  magnitude  in  the  proper  place  for 
Neptune  being  recorded  in   their  star-catalogues,  while  the  place   is  now 
vacant.     These  old  observations,  thus  recovered,  were  of  great  use  in  deter- 
mining the  planet's  orbit  with  accuracy. 

657.  The  Orbit  of  Neptune. — The  planet's   mean  distance  from 
the  sun  is  a  little  more  than  2800,000000  of  miles,  instead  of  being 
over  3600,000000,   as  it  should  be  according  to  Bode's  law.     The 
orbit,  instead  of  being  considerably  eccentric,  as  it  appeared  to  be 
from  the  computation  of  Adams  and  Leverrier,  is  more  nearly  circular 
than  any  other  in  the  system  except  that  of  Venus,  its  eccentricity 
being  only  y^g-p.     Even  this  small  fraction,  however,  makes  a  varia- 
tion of  over  50,000000  of  miles  in  the  planet's  distance  from  the  sun 
at  different  parts  of  its  orbit.     The  inclination  of  the  orbit  is  about 
lf°.     The  period  of  the  planet  is  about  164  years,  instead  of  217,  as 
it  should  have  been  according  to  Leverrier's  computed  orbit.     The 
orbital  velocity  is  about  3^  miles  a  second. 

658.  The  Solar  System  as  seen  from  Neptune. — At  Neptune's 
distance  the  sun  itself  has  an  apparent  diameter  of  only  a  little  more 
than  1'  of  arc,  — only  about  the  diameter  of  Venus  when  nearest  us, 
and  too  small  to  be  seen  as  a  disc  by  the  eye,  if  there  are  eyes  on 
Neptune.     The  light  and  heat  received  from  it  are  only  -^  part  of 


NEPTUNE.  403 

what  we  get  at  the  earth.     Still,  we  must  not  imagine  that,  as  com- 
pared with  starlight  or  even  moonlight,  the  Neptunian  sunlight  is  feeble. 

Assuming  Zb'llner's  estimate  that  sunlight  at  the  earth  is  618,000  times  as 
intense  as  the  light  of  the  full  moon,  we  find  that  the  sun,  even  at  Neptune, 
gives  a  light  equal  to  687  full  moons.  This  is  about  seventy-eight  times  the 
light  of  a  standard  candle  at  one  metre  distance,  or  about  the  light  of  a 
thousand  candle  power  electric  arc  at  a  distance  of  10 1  feet  —  abundant  for 
all  visual  purposes.  In  fact,  as  seen  from  Neptune,  the  sun  would  look  very 
much  like  a  large  electric  arc-lamp  at  a  distance  of  a  few  feet.  We  call 
special  attention  to  this,  because  erroneous  statements  are  not  unfrequently 
met  with  that  "  at  Neptune  the  sun  would  be  only  a  first  magnitude  star." 
It  would  really  give  about  44,000000  times  the  light  of  such  a  star. 

659.  From  Neptune  the  four  terrestrial  planets  would  be  hopelessly 
invisible,  unless  with   powerful  telescopes   and  by  carefully  screening  off 
the  sunlight.     Mars  would  never  reach  an  elongation  of  3°  from  the  sun ; 
the  maximum  elongation  of  the  earth  would  be  about  2°,  and  that  of  Venus 
about  1^°.     Jupiter,  attaining  an  elongation  of  about  10°,  would  possibly 
be  visible  in  the  twilight.     Neither  Saturn  nor  Uranus  would  be  conspicu- 
ous, the  latter  being  the  only  planet  of  the  whole  system  that  can  be  better 
seen  from  Neptune  than  it  can  be  from  the  earth. 

660.  The  Planet  itself.  —  Neptune  appears  in  the  telescope  as  a 
small  star  of  between  the  eighth  and  ninth  magnitudes,  absolutely 
invisible  to  the  naked  eye,  though  easily  seen  with  a  good  opera-glass. 
It  shows  a  greenish  disc,  having  an  apparent  diameter  of  about  2". 6, 
which  varies  very  little,  since  the  entire  range  of  variation  in  the 
planet's  distance  from  us  is  only  about  -^  of  the  whole.    The  real 
diameter  of  the  planet  is  about  35, OOO1  miles  (but  the  probable  error 
of  this  must  be  full}7  500  miles) ;  the  volume  is  about  85  times  that 
of  the  earth.     Its  mass,  as  determined  by  means  of  its  satellite,  is 
about  17  times  that  of  the  earth,  and  its  density  0.200 

The  planet's  albedo,  according  to  Zollner,  is  about  forty-six  per 
cent,  a  trifle  lower  than  that  of  Saturn  and  Venus,  and  considerably 
below  that  of  Jupiter  and  Uranus.  There  are  no  visible  markings 
upon  its  surface,  and  nothing  is  known  as  to  its  rotation.  The 
spectrum  of  the  planet  appears  to  be  precisely  like  that  of  Uranus. 
The  light  is  so  feeble  that  the  ordinary  lines  of  the  solar  spectrum 
are  difficult  to  make  out,  but  there  are  a  number  of  heavy,  dark 
bands,  which  indicate  the  existence  of  a  dense  atmosphere,  through 
which  the  light,  reflected  by  the  cloud-covered  surface  of  the  planet, 
is  transmitted,  —  an  atmosphere  which  appears  to  be  identically  the 

1  See  second  note  on  page  406. 


404  NEPTUNE'S  SATELLITE. 

same  on  .both  Uranus  and  Neptune,  while  some  of  its  constituents  are 
probably  present  in  Jupiter  and  Saturn,  as  shown  by  the  principal 
dark  band  in  the  red.  It  is  not  possible  as  yet  to  identifj'  the 
substance  which  produces  these  bands. 

It  will  be  seen  that  Uranus  and  Neptune  form  a  "  pair  of  twins  " 
very  much  as  the  earth  and  Venus  do ;  being  nearly  alike  in  magni- 
tude, density,  and  other  characteristics. 

661.  Satellite.  —  Neptune  has  one  satellite,  discovered  by  Lassell 
within  a  month  after  the  discovery  of  the  planet  itself.     Its  distance 
is  223,000  miles,  and  its  period  is  5d  21h  2m.7.     Its  orbit  is  inclined 
34°  53',  and  it  moves  backward   in  it;    i.e.,   clockwise,   from   east 
towards  the  west,  like  the  satellites  of  Uranus.     It  is  a  very  small 
object,  appearing  of  about  the  same  brightness  as  Oberon,  the  outer 
satellite  of   Uranus.     From  its  brightness,   as   compared  with   that 
of  Neptune  itself,  we  estimate  that  its  diameter  is  about  the  same  as 
that  of  our  own  moon,  though  perhaps  a  little  larger. 

662.  Trans-Neptunian  Planet. — Perhaps  the  breaking  down  of  Bode's 
law  at  Neptune  may  be  regarded  as  an  indication  that  the  system  terminates 
with  him,  and  that  there  is  no  remoter  planet.     If  such  a  planet  exists,  how- 
ever, it  is  sure  to  be  found  sooner  or  later,  either  by  means  of  its  disturbing 
action  upon  Uranus  and  Neptune,  or  else  by  the  methods  of  the  asteroid 
hunters,  although,  of  course,  its  slow  motion  will  render  its  detection  in  this 
way  difficult.     Several  observers  have  already  devoted  a  good  deal  of  time 
and  labor  to  the  search. 

663.  In  the  Appendix,  we  give  tables  containing  the  most  accurate  data 
of  the  planetary  system  at  present  available,  but  with  renewed  cautions  to 
the  student  that  these  data  are  of  very  different  degrees  of  accuracy.  % 

The  distances  (in  astronomical  units),  and  the  periods  of  the  planets 
(except  perhaps  some  of  the  asteroids)  are  known  with  extreme  precision ; 
probably  the  very  last  figure  of  the  table  may  be  trusted.  The  other 
elements  of  their  orbits  are  also  known  very  closely,  if  not  quite  so  precisely 
as  the  distances  and  periods.  The  masses,  in  terms  of  the  sun's  mass,  stand 
next  to  the  orbit-elements  in  order  of  precision,  with  an  error  probably  not 
exceeding  one  per  cent  (except,  however,  in  the  case  of  Mercury,  the  mass 
of  which  remains  still  very  uncertain). 

The  ratio  of  the  earth's  mass  to  the  sun's  is  however  less  accurately  known, 
being  at  present  subject  to  an  uncertainty  of  at  least  one  per  cent.  This  is 
because  its  determination  involves  a  knowledge  of  the  solar  parallax  (Art. 
278*),  the  cube  of  which  appears  in  the  formula  for  the  ratio  of  the  masses. 

Of  course  all  the  masses  of  the  planets  expressed  in  terms  of  the  earth's  mass  are 
subject  to  the  same  uncertainty  in  addition  to  all  other  possible  causes  of  error. 


THE    PLANETS.  405 

When  we  come  to  the  diameters,  volumes,  and  densities  of  the  planets,  the 
data  become  less  and  less  certain,  as  has  been  pointed  out  before.  For  the 
nearer  and  larger  planets,  say  Venus,  Mars,  and  Jupiter,  they  are  reason- 
ably satisfactory,  for  the  remoter  ones  less  so,  and  the  figures  for  the  density 
of  the  distant  planets,  —  Mercury,  Uranus,  and  Neptune,  for  instance,  —  are 
very  likely  subject  to  errors  of  ten  or  twenty  per  cent,  if  not  more. 

664.  We  borrow  from  HerschePs  "  Outlines  of  Astronomy  "  the  following 
illustration  of  the  relative  magnitudes  and  distances  of  the  members  of  our 
system.  "Choose  any  well-levelled  field.  On  it  place  a  globe  two  feet  in 
diameter.  This  will  represent  the  sun ;  Mercury  will  be  represented  by  a 
grain  of  mustard-seed  on  the  circumference  of  a  circle  164  feet  in  diameter 
for  its  orbit ;  Venus,  a  pea  on  a  circle  of  284  feet  in  diameter ;  the  Earth  also, 
a  pea  on  a  circle  of  430  feet ;  Mars,  a  rather  large  pin's-head  on  a  circle  of 
654  feet ;  the  asteroids,  grains  of  sand  in  orbits  of  1000  to  1200  feet ;  Jupiter, 
a  moderate-sized  orange  in  a  circle  nearly  half  a  mile  across ;  Saturn,  a  small 
orange  on  a  circle  of  four-fifths  of  a  mile ;  Uranus,  a  full-sized  cherry  or 
small  plum  upon  the  circumference  of  a  circle  more  than  a  mile  and  a  half ; 
and  finally  Neptune,  a  good-sized  plum  on  a  circle  about  two  miles  and  a  half 
in  diameter.  ...  To  imitate  the  motions  of  the  planets  in  the  above-men- 
tioned orbits,  Mercury  must  describe  its  own  diameter  in  41  seconds ;  Venus, 
in  4m  14s ;  the  Earth,  in  7  minutes ;  Mars,  in  4m  488 ;  Jupiter,  in  2h  56m  ; 
Saturn,  in  3h  13m ;  Uranus,  in  2h  16™ ';  and  Neptune,  in  3h  30m."  We  may 
add  that  on  this  scale  the  nearest  star  would  be  on  the  opposite  side  of  the 
globe,  at  the  antipodes,  8000  miles  away. 


>c  EXERCISES  ON  CHAPTER  XVI. 

*  1.  When  Jupiter  is  visible  in  the  evening,  do  the  shadows  of  his  satellites 
precede  or  follow  the  satellites  as  they  cross  the  planet's  disc  ? 

I       sJ^ 

*  2.   On  which  limb,  the  eastern  or  the  western,  do  the  satellites  appear  to 
enter  upon  the  disc  ? 

*  3.   What  probable  effect  would  the  great  mass  of  Jupiter  have  upon  the 
size  of  animals  inhabiting  it,  if  there  were  any  ? 

4.  How  would  sunlight  upon  Saturn  compare  with  sunlight  on  the  earth  ? 
How  with  moonlight  ? 

5.  What  would  be  the  greatest  elongation  of  the  earth  from  the  sun  as 
seen  from  Jupiter  ;  from  Saturn  ;  from  Uranus  ? 


406  EXERCISES. 

6.    What  would  be  the  apparent  angular  diameter  of  the  earth  when 
transiting "  the  sun  as  seen  from  Jupiter  ? 

•  7.  What  is  the  rate  in  miles  per  hour  at  which  a  white  spot  on  the 
equator  of  Jupiter,  showing  a  rotation-period  of  9h  50m,  would  pass  a  dark 
spot  indicating  a  period  of  9h  55m  ? 

NOTE  TO  ARTS.  631  AND  643. 

THE  NEW  SATELLITES.  The  sixth  and  seventh  satellites  of  Jupiter  were 
discovered  in  January  and  February,  1905,  by  Perrine,  at  the  Lick  Observatory, 
on  photographs  made  with  the  Crossley  reflector.  They  are  both  extremely 
small, — the  seventh  the  smaller, — and  probably  beyond  the  reach  of  visual 
observation.  They  are  far  outside  the  region  of  the  older  satellites,  —  a  pair  of 
twins  with  orbits  of  nearly  the  same  size,  more  than  seven  million  miles  in 
diameter,  inclined  about  30°  to  the  plane  of  the  planet's  equator  and  to  each 
other.  But  the  data  given  in  Table  II  are  likely  to  be  modified  by  later 
observations. 

Phcebe,  the  ninth  satellite  of  Saturn,  was  first  announced  by  Professor  W.  H. 
Pickering,  in  1898,  as  found  on  photographs  made  at  Arequipa  with  the  Bruce 
telescope.  The  discovery  remained,  however,  without  confirmation  until  1904, 
when  the  satellite  was  again  found  upon  a  large  number  of  later  photographs, 
sufficient  to  permit  a  reasonably  accurate  determination  of  its  orbit.  The  dis- 
tance from  the  planet  is  about  8  000000  miles,  the  period  18  months,  and  the 
orbital  motion  is  retrograde ! 

Themis,  Saturn's  tenth  satellite,  was  found  by  Pickering  in  April,  1905,  upon 
nine  of  the  plates  which  had  been  used  in  the  investigation  of  Phcebe.  She  is 
a  little  twin  sister  of  Hyperion,  but  is  three  magnitudes  fainter,  and  has  an  orbit 
of  almost  the  same  size  and  period,  though  more  eccentric  and  differently  tilted. 
The  data  given  in  Table  II  are  to  be  regarded  as  provisional.  It  is  worth  noting 
that  a  number  of  the  plates  which  were  examined  with  reference  to  Themis  show 
unexplained  objects,  —  possibly  asteroids,  possibly  other  satellites. 

NOTE  TO  ARTS.  647  AND  660. 

Professor  See  at  the  U.  S.  Naval  Observatory  found  much  smaller  values  for 
the  diameters  of  Uranus  and  Neptune.  For  Uranus  he  got  27,930  miles,  and 
for  the  latter  27,190,  reversing  the  hitherto  received  order  of  magnitude.  This 
illustrates  very  well  the  uncertainty  still  hanging  about  the  determination  of 
the  diameter  of  a  small  luminous  disc. 


DETERMINATION    OF   THE   SUN'S   DISTANCE.  407 


CHAPTER  XVII. 

THE  DETERMINATION  OF  THE  SUN'S  HORIZONTAL  PARALLAX 
AND  DISTANCE.  —  TRANSITS  OF  VENUS  AND  OPPOSITIONS  OF 
MARS.  —  GRAVITATIONAL  METHODS.  —  DETERMINATION  BY 
MEANS  OF  THE  VELOCITY  OF  LIGHT. 

665.  THIS  problem,  from  some  points  of  views,  is  the  most  funda- 
mental of  all  that  are  encountered  by  the  astronomer.    It  is  true  that 
it  is  possible  to  deal  with  many  of  the  subjects  that  present  them- 
selves in  the  science  without  the  necessity  of  employing  any  units 
of  length  and  mass  but  those  that  are  purely  astronomical,  leaving 
for  subsequent  determination  the  relation  between  these  units  and 
the  more  familiar  ones  of  ordinary  life  :  we  can  get,  so  to  speak,  a 
map  of  the  solar  system,  correct  in  proportion,  though  without  a  scale 
of  miles.     But  to  give  the  map  its  real  meaning  and  use,  we  must 
find  the  scale  finally,  if  not  at  first,  and  until  this  is  done  we  can 
form  no  true  conceptions  of  the  actual  dimensions,  masses,  and 
distances  of  the  heavenly  bodies. 

The  problem  of  finding  the  true  value  of  the  astronomical  unit  is 
difficult,  because  of  the  great  disproportion  between  the  size  of  the 
earth  and  the  distance  of  the  sun.  The  relative  smallness  of  the 
earth  limits  the  length  of  our  available  "base  line,"  which  is  less 
than  T^i<j^  part  of  the  distance  to  be  determined  by  it.'  It  is  as  if 
a  person  confined  in  an  upper  room  with  a  wide  prospect  were  set  to 
determine  the  distance  of  objects  ten  miles  or  more  away,  without 
going  outside  the  limits  of  his  single  window.  It  is  hopeless  to  look 
for  accurate  results  by  direct  methods,  such  as  answer  well  enough 
in  the  moon's  case,  and  astronomers  are  driven  to  indirect  ones.  • 

666.  Historical.  —  Until   nearly  1700   no   even  reasonably  accurate 
knowledge  of  the  sun's  distance  had  been  obtained.     Aristarchus,  by  a  very 
ingenious  though  inaccurate  method,  had  found,  as  he  thought,  that  the 
distance  of  the  sun  was  nineteen  times  as  great  as  that  of  the  moon  (it  is 
really  390  times  as  great),  and  Hipparchus,  combining  this  determination  of 
Aristarchus  with  his  own  knowledge  of  the  moon's  distance,  estimated  the 


408  DETERMINATION   OF   THE   SUN'S   DISTANCE. 

sun's  parallax  at  3',  which  is  more  than  twenty  times  too  large.  This  value 
was  accepted  by  Ptolemy,  and  remained  undisputed  for  twelve  centuries,  until 
Kepler,  from  Tycho's  observations  of  Mars,  satisfied  himself  that  the  sun's 
parallax  could  not  exceed  1' ;  i.e.,  that  the  sun's  distance  must  be  at  least  as 
great  as  twelve  or  fifteen  millions  of  miles.  Between  1670  and  1680  Cassini 
proposed  to  determine  the  solar  parallax  by  observations  of  Mars ;  for  by  that 
time  the  distance  of  Mars  from  the  earth  at  any  moment  could  be  very  accu- 
rately computed  in  astronomical  units,  so  that  the  determination  of  the  par- 
allax of  Mars  would  make  known  that  of  the  sun.  Observations  in  France, 
compared  with  observations  made  in  South  America  by  astronomers  sent  out 
for  the  purpose,  showed  that  the  parallax  of  Mars  could  not  exceed  25",  or 
that  of  the  sun,  10".  Cassini  set  it  at  9".5,  corresponding  to  a  distance  of 
86,000000  of  miles,  —  giving  the  first  reasonable  approach  to  the  true  dimen- 
sions of  the  solar  system. 

In  1677,  and  more  fully  in  1716,  Halley  explained  how  transits  of  Venus 
might  be  utilized  to  furnish  a  far  move  accurate  determination  of  the  solar 
parallax  than  was  possible  by  any  method  before  used.  He  died  before  the 
transits  of  1761  and  1769  occurred,  but  they  were  both  observed,  the  first 
not  very  satisfactorily,  but  the  second  with  perfect  success,  and  in  the  most 
widely  separated  parts  of  the  globe.  The  results,  however,  were  by  no 
means  as  accordant  as  had  been  expected.  Various  values  of  the  sun's  par- 
allax were  deduced,  ranging  all  the  way  from  8}"  to  9",  according  to  the 
observations  used,  and  the  way  they  were  treated  in  the  discussion. 
Towards  the  end  of  the  century,  La  Place  adopted  and  used  for  a  while  the 
value  8".81,  while  Delambre  preferred  8".6. 

667.  In  1822-24  Encke  collected  all  the  transit  observations  that 
had  been  published,  and  discussed  them  as  a  whole  in  an  extremely 
thorough  manner,  deducing  as  a  final  result  from  the  two  transits  of 
1761  and  1769,  8". 5776,  corresponding  to  a  distance  of  95 £  millions 
of  miles.  The  decimal  is  very  imposing,  and  the  discussion  by 
which  it  was  obtained  was  unquestionably  thorough  and  laborious, 
go  that  his  value  stood  unquestioned  and  classical  for  many  years. 

The  first  note  of  dissent  was  heard  in  1854,  when  Hansen,  in 
publishing  certain  researches  upon  the  motion  of  the  moon,  an- 
nounced that  Encke's  parallax  was  certainly  too  small ;  he  after- 
wards fixed  the  figure  at  8". 97,  but  the  correction  of  certain  numeri- 
cal errors  in  his  work  reduced  this  result  to  8". 92. 

Three  or  four  years  later  Leverrier  found  a  value  of  8".95  from  the  so- 
called  lunar  equation  of  the  sun's  motion;  and  in  1862  Foucault,  from  a  new 
determination  of  the  velocity  of  light,  combined  with  the  constant  of  aber- 
ration, got  the  value  8". 86.  A  re-discussion  of  the  old  transit  of  Venus 
observations  was  then  made  by  Stone,  of  England,  who  deduced  from  them 


METHODS    OF   FINDING   THE   SOLAR    PAKALLAX.  409 

a  value  of  8".943.  The  value  of  8".9o  was  adopted  by  the  British  Nautical 
Almanac,  and  used  in  it  until  the  issue  of  1882.  The  corresponding  dis- 
tance of  91^  millions  of  miles  found  its  way  into  numerous  text-books,  and, 
though  known  to  be  erroneous,  still  holds  its  place  in  some  of  them. 

In  1867  Newcomb  made  a  discussion  of  all  the  data  then  avail- 
able, and  obtained  the  value  8 ".848  (or  8 ".85  practically),  which 
value  is  still  (1897)  used  in  all  the  Nautical  Almanacs  except  the 
French,  which  uses  8  ".86.  After  1900,  however,  it  is  agreed  to  use 
8".80  in  all  of  them. 

668,  The  observations  of  Gill  on  the  planet  Mars  in  1877,  and 
the  new  determinations  of  the  velocity  of  light  by  Michelson  and 
Newcomb  in  this  country,  as  well  as  the  investigations  of  Neison 
and  others  upon  the  so-called  "parallactic  inequality"  of  the  moon, 
all  point,  however,  to  a  somewhat  smaller  value.     Professor  New- 
comb  says  (in  1885),  "All  we  can  say  at  present  is  that  the  solar 
parallax  is  probably  between  8 ".78  and  8  ".83,  or  if  outside  these 
limits,  that  it  can  be  very  little  outside."     The  latest  investigations 
fully  confirm  this  conclusion.     (See  note  at  end  of  the  chapter.) 

It  was  not,  however,  thought  worth  while  to  change  the  constant  used  in 
the  almanacs  until  the  final  reduction  of  the  transits  of  1874  and  1882  had 
been  made,  and  until  certain  experiments  and  investigations  in  progress 
have  been  finished.  The  difference  between  8".80  and  8".85  is  of  no  prac- 
tical account  for  almanac  purposes,  and  the  change  would  involve  alterations 
in  a  number  of  the  tables. 

Accepting  Clarke's  value  of  the  earth's  equatorial  radius  (Art.  145),  viz., 
6  378206.4m  or  3963.3  miles,  we  find  that  a  solar  parallax  of 

8".75  corresponds  to  23573  radii  of  the  earth  =  93428000  miles. 
8".80  "  "  23439     "      "     "       "     =92897000      " 

8".85  "  "  23307     "      "    "       "     =  92372000      " 

8".90  "  "  23196     «      "    "       "      =91852000      " 

669.  Methods  of  finding  the  Solar  Parallax  and  Distance.  —  We 

may  classify  them  as  follows  :  — 

I.  Ancient  Methods. 

(A)  Method  of  Aristarchus  [0]. 

(B)  Method  of  Hipparchus  [0]. 

II.  Geometrical  and  Trigonometrical  Methods,  in  which  we  attempt 
to  find  by  angular  measurements  the  parallax,  either  of  the  sun 
itself  or  of  one  of  the  nearer  planets. 


410  DETERMINATION  OF   THE  SUN'S  DISTANCE. 


The  direct  method  [0]. 
(B)  Observations  of  the  displacement  of  Mars  among  the  stars 
at  the  time  of  opposition. 

(a)  Declination  observations  from  two  or  more  stations  in  widely 
different  latitudes  made  with  meridian  circles  or  micrometer 
[25]. 

(6)  Observations  made  at  a  single  station  near  the  equator,  by 
measuring  the  distance  of  the  planet  east  or  west  from 
neighboring  stars,  using  the  heliometer  [90]. 

(O)  Declination  observations  of  Venus  [20]. 

(D)  Observations  of  some  of  the  nearer  asteroids  in  the   same 
way  as  Mars. 

(a)  Meridian  observations  at  two  stations  in  widely  different 

latitudes  [20]. 
(6)  Heliometer  observations  at  an  equatorial  station  [90]. 

(E)  Observations  of  the  transits  of  Venus  at  widely  separated 
stations. 

(a)  Observations  of  the  contacts. 

(1)  Halley's  method  —  the  "method  of  durations"  [40]. 

(2)  Delisle's  method  —  observation  of  absolute  times  [50]. 

(6)  Heliometer  measurements  of  the  position  of  the  planet  on 

the  sun  [75]. 
(c)  Photographic  methods  —  various  [20  to  75], 

III.  Gravitational  Methods. 

(A)  By  the  moon's  parallactic  inequality  [70]. 

(B)  By  the  lunar  equation  of  the  sun's  motion  [40]. 

(O)  By  the  perturbations  produced  by  the  earth  on  Venus  and 
Mars  [70]  ;   (ultimately  [95]). 

IV.  By  the  Velocity  of  Light,  combined  with 

(A)  The  light  equation  [80]. 

(B)  The  constant  of  aberration  [90].1 

The  figures  in  brackets  at  the  right  are  intended  to  express  roughly  the 
relative  value  of  the  different  methods,  on  the  scale  of  100  for  a  method 
which  would  insure  absolute  accuracy. 

670.  Of  the  Ancient  Methods,  that  of  Aristarchus  is  so  ingenious 
and  simple  that  it  really  deserved  to  be  successful.  When  the  moon 
is  exactly  at  the  half  phase,  the  angle  at  M  (Fig.  184)  must  be  just 

1  For  spectroscopic  method,  see  note  on  page  427. 


GEOMETRICAL   METHODS.  411 

90°,  and  the  angle  AEM  must  equal  MSE.     If,  then,  we  can  find 

how  much  shorter  the  arc  NM  (from  new  to  half  moon)  is  than  MF 

(from  half  moon  to  full),  half  the  difference  will  measure  AM,  and 

give  the  angle  at  S.     Aristarchus  concluded  that  ths  first  quarter  of 

the  month  was  just  about  twelve  hours  shorter  than  the  second,  so 

that  AM  was  measured  by 

six  hours'  motion  of  the 

moon,  or  a  little  less  than 

4°.     Hence  he  found  SE, 

the  distance  of  the  sun, 

to  be  about  nineteen  times 

EM  —  a  value   absurdly  FIG  Jg4 

Wrong,  Since  SE  is  in  fact      Aristarchus,Methodof  Determining  the  Sun's  Distance. 

nearly  390  times  EM.  The 

real  difference  between  the  two  quarters  of  the  month  is  only  about 

half  an  hour,  instead  of  twelve  hours. 

The  difficulty  with  the  method  is  that,  owing  to  the  ragged  and 
broken  character  of  the  lunar  surface,  it  is  impossible  to  observe 
the  instant  of  half  moon  with  sufficient  accuracy. 

671.  The  estimate  of  Hipparchus  was  based  upon  the  erroneous  calcu- 
lation of  Aristarchus  that  the  sun's  distance  is  19  times  the  moon's,  and  the 
solar  parallax,  therefore,  ^  of  the  moon's  parallax. 

The  "  radius  of  the  earth's  shadow,"  where  the  moon  cuts  it  at  a  lunar 
eclipse,  is  given,  as  Hipparchus  knew,  by  the  formula  p  =  P  +  p  —  S  (Art. 
372),  or  P  +  p  =  p  +  S.  Assuming  that  P  =  Wp,  we  have  2Qp  =  p+S.  Now 
S,  the  sun's  semi-diameter,  is  about  15';  and  from  the  duration  of  lunar 
eclipses  Hipparchus  found  p  to  be  about  40' ;  hence  he  obtained  for  p,  the 
solar  parallax,  a  value  a  little  less  than  3',  which,  as  has  been  already  men- 
tioned, was  accepted  by  Ptolemy,  and  by  succeeding  astronomers  for  more 
than  1500  years.  (Wolf's  "  History  of  Astronomy,"  p.  175.) 

672.  Of  the  Geometrical  Methods,  A,  the  "direct  method7'  con- 
sists in  observing  the  sun's  apparent  declination  with  the  meridian 
circle  at  two  stations  widely  differing  in  latitude,  just  as  we  observe 
the   moon  when   finding  its   parallax   (Art.   239).      Theoretically, 
observations  of  this  sort  might  give  the  value  of  the  sun's  parallax 
within  -J-"  or  so,  but  the  method  is  practically  worthless,  because 
the  errors  of  observation  are  large  as  compared  with  the  quantity 
to  be  determined.     The  sun's  limb  is  a  very  bad  object  to  point  on, 
and  besides,  its  heat  disturbs  the  adjustments  of  the  instrument,  thus 
rendering  the  observations  still  more  inaccurate. 


412 

673.  The  first  of  the  two  methods  of  observing  the  planet  Mars 
is  precisely  the  same  as  this  direct  method  of   observing  the   sun; 
but   the   distance  of   Mars  at  a  "near  opposition"   is  only  a  little 
more    than  ^  that   of   the    sun,    so   that   any    error   of   observation 
affects    the    final   result  by  only  about  J  as  much ;    and,  moreover, 
Mars  is  a  very  good  object  to  observe,  so  that  the  errors  of  observa- 
tion   themselves    are    much   lessened.      The   planet's    distance  from 
the  earth   having  been  found  in  astronomical  units  by  the  method 
of  Art.  515,  the  determination  of  its  distance  in  miles  will  fix  the 
value  of  this  unit,  and  so  give  us  directly  the  sun's    distance    and 
parallax. 

The  method  requires  two  observers  working  at  a  distance  from  each 
other  with  different  instruments,  which  is  a  serious  disadvantage. 

For  some  unexplained  reason,  observations  of  this  sort  seem  almost  inva- 
riably to  give  too  large  a  result  for  the  solar  parallax,  averaging  between 
8". 90  and  8".98.  The  red  color  of  the  planet  may  possibly  have  something 
to  do  with  this  by  affecting  the  astronomical  refraction.  This  method,  in 
1680,  was  the  first  to  give  a  reasonable  approximation  to  the  sun's  true 
distance,  as  has  been  mentioned  before. 

The  planet  Venus  can  be  observed  in  the  same  way,  and  has  been  once 
so  observed  by  Gillis,  1849-52,  at  Santiago,  Chili,  in  co-operation  with  the 
Washington  observers,  but  the  result  was  not  very  satisfactory. 

674.  Heliometer  Observations  of  Mars  (Method  b) .  —  It  is  pos- 
sible, however,  for  a  single  observer  to  obtain   better   results   than 
can  be  got  by  two  or  more  using  the  preceding  method.     Suppose 
that  the  orbital  motion  of  Mars  is  suspended  for  a  while  at  oppo- 
sition,  and  that   the   planet   is   on   or   near   the   celestial   equator ; 


FIG.  185.  — Effect  of  Parallax  on  the  Right  Ascension  of  Mars. 

and  also  that  the  observer  is  at  a  station,  0,  on  the  earth's 
equator.  When  Mars  is  rising  at  Met  Fig.  185,  the  horizontal 
parallax  OMeC  depresses  the  planet ;  that  is,  he  appears  from  0  to 
be  further  east  than  he  would  if  seen  from  C,  the  centre  of  the 
earth ;  so  that  the  parallax  then  increases  the  planet's  right  ascen- 
sion. Twelve  hours  later,  when  he  is  setting,  the  parallax  will 


HELIOMETER   OBSERVATIONS   OF   MARS. 


413 


w 


throw  him  towards  the  west,  diminishing  his  right  ascension  by  the 
same  amount.  If,  then,  when  the  planet  is  rising,  we  measure  care- 
fully its  distance  west  of  a  star  S,  which  is  supposed  to  be  just  east 
of  it  (the  distance  MeS  in  Fig.  186),  and  then  measure  the  distance 
MJS  from  the  same  star  again  when  it  is  setting,  the  difference  will 
give  us  twice  the  horizontal  parallax.  The  earth's  rotation  will 
have  performed  for  the  observer  the  function  of  a  long  journey  in 
transporting  him  from  one  station  to  another  8000  miles  away  in  a 
straight  line. 

675.  Of  course  the  observations  are  not  practically  limited  to  the 
moment  when  the  planet  is  just  rising,  nor  is  it  necessary  that  the 
star  measured  from  should  be  exactly  east  or  west  of  the  planet. 
Measures  from  a  number  of  the 
neighboring  stars,  S^  S2,  £3,  and 
$4  would  fix  the  positions  Me  and 
Mw  with  more  accuracy  than  meas- 
ures from  S  alone.  Nor  will  the 
planet  stop  in  its  orbit  to  be  ob- 
served, nor  will  it  have  a  declina- 
tion of  zero,  nor  can  the  observer  S 
command  a  station  exactly  on  the 
earth's  equator.  But  these  varia- 
tions from  the  ideal  conditions  do 
not  at  all  affect  the  principles  in- 
volved ;  they  simply  complicate  the 
calculations  slightly  without  com- 
promising their  accuracy. 

The  method  has  the  very  great 


*s, 

.        G 

K. 

s,    p 
\ 

Me 

' 

TV 

u 

s 

* 

04 

j; 

FIG.  186. 

Micrometric  Comparison  of  Mars  with  Neigh- 
boring Stars. 


advantage  that  all  the  observations 

are  made  by  one  person,  and  with 

one  instrument,  so  that,  as  far  as  can  be  seen,  all  errors  that  could 

affect  the  result  are  very  thoroughly  eliminated. 

676.  The  most  elaborate  determination  of  the  solar  parallax  yet  made 
by  this  method  is  that  of  Mr.  Gill,  who  was  sent  out  for  the  purpose  by  the 
Royal  Astronomical  Society  in  1877  to  Ascension  Island  in  the  Atlantic 
Ocean.  His  result,  from  350  sets  of  measurements,  gives  a  solar  parallax 
of  8".783  ±  0".015,  —  a  result  probably  very  close  to  the  truth,  though  pos- 
sibly a  little  small.  In  1892  and  1894  favorable  oppositions  of  Mars  again 
occurred,  and  the  observations  were  repeated  at  the  Cape  of  Good  Hope  and 
elsewhere  with  confirmatory  results. 


414  DETERMINATION    OF    THE    SUN'S    DISTANCE. 

Venus  cannot  be  observed  in  this  way,  since  either  her  rising  or  setting 
is  in  the  daytime,  when  the  small  stars  cannot  be  seen  near  her. 

676*.  Heliometer  Observations  of  Asteroids.  —  Heliometer  ob- 
servations of  the  nearer  asteroids  furnish  perhaps  the  best  of  all 
the  geometrical  methods.  It  is  true  that  the  minor  planets  are 
more  distant  than  Mars  (Eros  excepted),  hence  a  given  error  in  their 
observation  entails  a  greater  error  in  the  final  result.  But  on  the 
other  hand,  they  can  be  observed  with  far  greater  precision,  because 
they  appear  as  mere  star-like  points,  in  brightness  and  color  just 
like  the  stars  around  them  which  serve  as  points  of  reference. 

The  method  has  been  applied  several  times,  most  recently1  in  1889-90, 
when  three  of  the  asteroids,  Victoria,  Iris,  and  Sappho,  were  concertedly, 
and  most  carefully,  observed  by  Dr.  Gill  at  the  Cape  of  Good  Hope,  Dr. 
Elkin  at  New  Haven,  and  two  or  three  observers  in  Europe.  The  observa- 
tions have  been  thoroughly  reduced,  and  the  results  are  very  accordant  and 
apparently  extremely  accurate.  Gill  obtains  from  them  for  the  parallax  of 
the  sun  8".802  ±  0".005. 

677.      The  Heliometer.  —  The  heliometer,  the  instrument  employed  in 
these  measures,  is  one  of  the  most  important  of  the  modern  instruments  of 
precision.    As  its  name  implies,  it  was  first  designed  to  measure  the  diameter 
of  the   sun,  but  it  is  now  used  to  measure   any  distance  ranging  from 
a  few  minutes  up  to  one  or  two  degrees,  which  it  does  with  the  same  accu- 
racy as  that  with  which  the  filar  micrometer  measures  distances  of  a  few 
seconds.     It  is  a  "  double  image  "  micrometer,  made  by  dividing  the  object- 
glass  of  a  telescope  along  its  diameter,  as  shown 
A.\    A.O     A$  in  Fig-  187.     The  two  halves  are  so  mounted  that 

they  can  slide  by  each  other  for  a  distance  of  three 
or  four  inches,  the  separation  of  the  centres  being 
accurately  measured  by  a  delicate  scale,  or  by  a 
micrometer  screw  operated  and  read  by  a  suitable 
arrangement  from  the  eye-end.     The  instrument 
is  mounted  equatorially  with  clock-work,  and  the 
tube  can  be  turned  in   its   cradle   so  as  to  make 
*      ®     0°    Q2         the  line  of  division  of  the  lenses  lie  in  any  desired 
Si     S0     Ss  direction.     When  the  centres  of  the  two  halves  of 

Fio.  187.— The  Heliometer.     the  object-glass  coincide,  the  whole  acts  as  a  single 
lens.     As  soon  as  the  centres  are  separated,  each 
half  of  the  object-glass  forms  its  own  image. 

To  measure  the  distance  from  Mars  to  a  star,  the  telescope  tube  is  turned 
so  that  the  line  of  centres  points  in  the  right  direction,  and  then  the  semi- 
lenses  are  separated  until  one  of  the  two  images  of  the  star  comes  exactly  in 
the  centre  of  one  of  the  images  of  Mars ;  this  can  be  done  in  two  positions 

1  See  note  on  page  426. 


H  ALLEY'S  METHOD.  415 

of  the  semi-lens  A  with  respect  to  B,  as  shown  by  the  figure.  We  may  either 
make  S0  (the  image  of  the  star  formed  by  semi-lens  .B)  coincide  with  M1 
formed  by  A,  or  make  S2  coincide  with  M0.  The  whole  distance  from  1  to  2 
then  measures  twice  the  distance  between  M  and  S. 

678.  Transit  of  Venus  Observations.  —  At  the  time  when  Venus 
passes  between  us  and  the  sun,  her  distance  from  the  earth  is  only 
some  26  000000  of  miles,  so  that  her  horizontal  parallax  is  nearly 
four  times  as  great  as  that  of  the  sun  itself.     At  this  time  her 
apparent  displacement  upon  the  sun's  disc,  due  to  a  change  of  the 
observer's  station  upon  the  earth,  is  the  difference  between  her  own 
parallax  due  to  this  displacement,  and  that  of  the  sun  itself ;  and 
this  difference  is  greater  than  the  sun's  parallax  nearly  in  the  ratio 
of  3  to  1,  or,  more  exactly,  of  723  to  277.1     The  object,  then,  of  the 
observations  of  a  transit  is  to  obtain  in  some  way  a  measure  of  the 
angular  displacement  of  Venus  on  the  sun's  disc,  corresponding  to 
the  known  distance  between  the  observer's  stations  upon  the  earth. 

679.  Halley's  Method.—  The  method  proposed  by  Halley,  who 
in  1677  brought  to  notice  the  great  advantages  presented  by  a  transit 
of   Venus    for    determining    the    sun's 

parallax,  was  as  follows :  Two  sta- 
tions are  chosen  upon  the  earth's  sur-  1  2/  ^- 
face,  as  far  separated  in  latitude  as 
possible.  From  them  we  observe  the 
duration  of  the  transit;  that  is,  the 
interval  of  time  between  its  beginning 
and  end,  both  of  which  must  be  visible 
at  both  stations.  If  the  clock  runs  cor- 
rectly during  the  f  e  w  h  ours  during  which 
the  transit  lasts,  this  is  all  that  is  neces- 

.  .  Contacts  in  a  Transit  of  Venus. 

sary.    We  do  not  need  to  know  its  error 

in  reference  to  Greenwich  time,  nor  even  in  respect  to  the  local  time, 
except  roughly.  This  was  a  great  advantage  of  the  method  in  those 
days,  before  the  era  of  chronometers,  when  the  determination  of  the 
longitude  of  a  place  was  a  very  difficult  and  uncertain  operation. 

1  Since  the  distance  of  Venus  from  the  sun  is  0.723  of  the  astronomical  unit, 
her  distance  from  the  earth  at  time  of  transit  is  0.277.  Let  p  be  the  sun's 
parallax  and  v  that  of  Venus :  then,  since  the  parallax  of  a  body  is  inversely 

1000 
proportional  to  its  distance  from  the  earth  (Art.  83),  v  =  p  X  i^p    and  v  —  p 

/ 1000- 277  \  _  ^  723 


/ 1000- 277  \  723 

(       277        )  =  *  *  277' 


416  DETERMINATION    OF    THE    SUN'S   DISTANCE. 

The  observation  to  be  made  is  simply  to  note  the  clock  time  at  which 
"contact"  occurs,  there  being  four  of  these  contacts,  —  two  exterior 
arid  two  internal,  at  the  points  marked  1,  2,  3,  4,  in  Fig.  188.  Halley 
depended  mainly  on  the  two  internal  contacts,  which  he  supposed 
could  be  observed  with  an  error  not  exceeding  one  second  of  time. 

680.  Computation  of  the  Parallax.  —  Having  the  durations  of  the 
transits  at  the  two  stations,  and  knowing  the  hourly  angular  motion 
of  Venus,  we  have  at  once  and  very  accurately  the  length  of  the  two 
chords  described  by  Venus  upon  the  sun,  expressed  in  seconds  of 
arc.  We  also  know  the  sun's  semi-diameter  in  seconds,  and  hence 


(Earth) 


FIG.  189.  —  Halley's  Method. 

in  the  triangles  (Fig.  189)  Sab  and  Sde,  we  can  compute  the  length 
(in  seconds  still)  of  Sb  and  Se,  the  difference  of  which,  be,  is  the 
angular  displacement,  due  to  the  distance  between  the  stations  on  the 
earth.1  The  virtual  base  line  is,  of  course,  not  the  distance  between 
B  and  E  as  a  straight  line,  because  that  line  is  not  perpendicular  to 
the  line  of  sight  from  the  earth  to  Venus,  nor  to  the  plane  of  the 
planet's  orbit,  but  the  true  value  to  be  used  is  easily  found.  Calling 
this  base  line  ft,  we  have 

*(£)©• 

r  being  the  radius  of  the  earth. 

The  rotation  of  the  earth,  of  course,  comes  in  to  shift  the  places 
of  E  and  B  during  the  transit,  but  this  can  easily  be  allowed  for ; 
and  if  the  stations  are  well  situated,  it  increases  the  difference 
between  the  two  durations,  and  increases  the  accuracy  of  the  result. 

1  In  orfler  that  the  method  may  be  practically  successful,  it  is  necessary  that 
the  transit  track  should  lie  near  the  edge  of  the  sun's  disc,  for  two  reasons.  It 
is  desirable  that  the  duration  should  not  be  more  than  three  or  four  hours,  while 
for  a  central  transit  it  lasts  eight  hours  (Art.  575).  Moreover,  if  the  two  chords 
were  near  the  centre  of  the  disc,  any  small  error  in  the  length  of  either  chord 
would  produce  a  great  error  in  the  computed  distance  between  them.  When 
they  lie  as  in  the  figure  (which  has  been  the  case  in  all  recent  transits),  the  reverse 
is  true :  a  considerable  error  in  the  observed  length  of  one  of  the  chords  affects 
their  computed  distance  only  slightly. 


DELISLE'S  METHOP. 


41T 


681.  The  Black  Drop.  —  Halley  expected,  as  has  been  said,  that 
it  would  be  possible  to  observe  the  instant  of  internal  contact  within 
a  single  second  of  time,  but  he  reckoned  with- 
out his  host.  At  the  transits  of  1761  and 
1769,  at  most  of  the  stations  the  planet  at 
the  time  of  internal  contact  showed  a  "  liga- 
ment" or  "  black  drop,"  like  Fig.  190,  instead 
of  presenting  the  appearance  of  a  round  disc 
neatly  touching  the  edge  of  the  sun  ;  and  the 
time  of  real  contact  was  thus  made  doubtful 
by  10"  or  15s. 


FIG.  190.  — The  Black  Drop. 


This  "ligament"  depends  upon  the  fact  that  the  optical  edge  of  the 
image  of  a  bright  body  is  not,  and  in  the  nature  of  things  cannot  be,  abso- 
lutely sharp  in  the  eye  or  in  the  telescope.  In  the  eye  itself  we  have 
irradiation.  In  the  telescope  we  have  the  difficulty  that  even  in  a  perfect 
instrument  the  image  of  a  luminous  point  or  line  has  a  certain  width  (which 
with  a  given  magnifying  power  is  less  for  a  large  instrument).  Moreover 
a  telescope  is  usually  more  or  less  imperfect,  and  practically  adds  other 
defects  of  definition,  so  that  whenever  the  limbs  of  two  objects  approach 
each  other  in  the  field  of  view 
of  a  telescope  we  have  more  or 
less  distortion  due  to  the  over- 
lapping of  the  two  "penum- 
bras of  imperfect  definition," 
—  the  same  sort  of  effect  that 
is  obtained  by  putting  the 
thumb  and  finger  in  contact, 
holding  them  up  within  two 
or  three  inches  of  the  eye  and 
then  separating  them :  as  they 
separate,  a  "black  ligament" 
will  be  seen  between  them. 

With  modern  telescopes,  and 
by  great  care  in  preventing  the 
sun's  image  from  being  too 
bright,  so  as  to  diminish  irradi- 
ation in  the  eye  as  far  as  pos- 
sible, the  black  drop  was  re- 
duced to  reasonably  small  pro- 
portions in  1874  and  1882,  and 

practice  beforehand  with  an  "  artificial  transit "  enabled  the  observer  in  some 
degree  to  allow  for  its  effect.  But  a  new  difficulty  appeared,  from  which 
there  seems  to  be  absolutely  no  way  of  escape,  —  the  planefs  atmosphere 


FIG.  191.  —Atmosphere  of  Venus  as  seen  during  a 
Transit.     (Vogel,  1882.) 


418  DETERMINATION    OF    THE    SUN'S    DISTANCE. 

causes  it  to  be  surrounded  by  a  luminous  ring  as  it  enters  upon  the  sun's 
disc,  and  thus  renders  the  time  of  the  contact  uncertain  by  at  least  five 
or  six  seconds.  In  both  the  transits  of  1874  and  1882,  differences  of  that 
amount  continually  appeared  among  the  results  of  the  best  observers.  Fig- 
191  shows  the  appearances  due  to  this  cause  as  observed  by  Vogel  in  1882. 

682.  Delisle's  Method. —  Halley '*s  method  requires  the  use  of  polar 
stations,  uncomfortable  and  hard  to  reach,  and  also  that  the  weather 
should  permit  the  observer  to  see  both  the  beginning  and  end  of  the 
transit. 

Delisle's  method,  on  the  other  hand,  utilizes  two  stations  near  the 
equator,  taken  on  a  line  roughly  parallel  to  the  planet's  motion.  It 
requires  also  that  the  observers  should  know  their  longitude  accurately, 
so  as  to  be  able  to  determine  the  Greenwich  time  at  any  moment ; 
but  it  does  not  require  that  they  should  see  both  the  beginning  and 
end  of  the  transit ;  observations  of  either  phase  can  be  utilized : 
and  this  is  a  great  advantage.  Suppose,  then  (Fig.  192),  that  the 
observer  W  on  one  side  of  the  earth  notes  the  moment  of  Internal 
contact  in  Greenwich  time,  the  planet  then  being  at  V\.  When  E 
notes  the  contact  (also  in  Greenwich  time)  the  planet  will  be  at  F2, 
and  the  angle  at  D  will  be  the  angular  diameter  of  the  earth  as  seen 


I) 


E 

Martti 


FIG.  192.  — Delisle's  Method. 

from  D ;  i.e.,  simply  twice  the  sun's  parallax.  Now  the  angle  D  is  at 
once  determined  by  the  time  occupied  by  Venus  in  passing  from  Fi 
to  F2,  since  in  584  days  (the  synodic  period)  she  moves  completely 
round  from  the  line  DWto  the  same  line  again.  If  the  time  from 
F!  to  F2  were  twelve  minutes,  we  should  find  the  angle  at  D  to  be 
about  18". 

The  observations  of  the  internal  contacts  of  the  transits  of  1874  and 
1882  give  results  according  to  Newcomb  ranging  from  8".72  to  8".88,  with 
a  weighted  mean  according  to  Newcomb  of  8".794. 

683.  Heliometer  Observations.  — Instead  of  observing  simply  the 
times  of  contact,  and  leaving  the  rest  of  the  transit  unutilized,  as  in 
the  two  preceding  methods,  it  is  possible  to  make  a  continuous  series 


TRANSIT  OF  VENUS  PHOTOGRAPHY.          419 

of  measurements  of  the  distance  and  direction  of  the  planet  from  the 
nearest  point  of  the  sun's  limb.  These  measurements  are  best  made 
with  the  heliometer  (Art.  677),  and  give  the  means  of  determining 
the  planet's  apparent  position  upon  the  sun's  disc  at  any  moment 
with  extreme  precision.  Such  sets  of  measurements,  made  at 
widely  separated  stations,  will  thus  furnish  accurate  determinations 
of  the  apparent  displacement  of  the  planet  on  the  sun's  disc,  corre- 
sponding to  known  distances  on  the  earth,  and  so  will  give  the  solar 
parallax. 

During  the  transit  of  1882  extensive  series  of  observations  of  this 
sort  were  made  by  the  German  parties,  two  of  which  were  in  the 
United  States,  —  one  at  Hartford,  Conn.,  and  the  other  at  San 
Antonio,  Texas. 

For  some  reason,  not  clearly  evident,  the  results,  while  very  accordant 
among  themselves,  are  considerably  larger  than  the  average  deduced  from 
other  methods.  From  the  heliometer  observations  of  1874  the  parallax 
came  out  8".876,  and  from  those  of  1882,  8".S79. 

684.  Photographic  Observations.  —  The  heliometer  measurements 
cannot  be  made  very  rapidly.     Under  the  most  favorable  circum- 
stances a  complete  set  requires  at  least  fifteen  minutes,  so  that  the 
whole  number   obtainable  during  the  seven  or  eight  hours   of  the 
transit  is  quite  limited.     Photographs,  on  the  other  hand,  can  be 
made  with  great  rapidity  (if  necessary,  at  the  rate  of  two  a  minute) , 
and  then  after  the  transit  we  can  measure  at  leisure  the  position  of 
the  planet  on  the  sun's  disc  as  shown  upon  the  plate.     At  first  sight 
this  method  appears  extremely  promising,  and  in  1874  great  reliance 
was  placed  upon  it.     Nearly  all  the  parties,  some  fifty  in  number, 
were   provided   with   elaborate   photographic   apparatus   of    various 
kinds.     On  the  whole,  however,  the  results,  upon  discussion,  appear 
to  be  no  more  accordant  than  those  obtained  by  other  methods,  so 
that  in  1882  the  method  was  generally  abandoned,  and  used  only  by 
the   American   parties,    who   employed   an   apparatus   having   some 
peculiar  advantages  of  its  own. 

685.  English,  German,  and  French  Methods.  — In  1874,  the  English 
parties  used  telescopes  of  six  or  seven  inches  aperture,  and  magnified  the 
image  of  the  sun  formed  by  the  object-glass  by  a  combination  of  lenses 
applied  at  the  eye-end.    There  were  no  special  appliances  for  eliminating  the 
distortion  produced  by  the  enlarging  lenses,  nor  for  ascertaining  the  exact 
orientation  of  the  picture  (that  is,  the  direction  of  the  image  upon  the  plate 
with  reference  to  north  and  south),  nor  for  determining  its  scale. 


420 


DETERMINATION    OF   THE   SUN'S   DISTANCE. 


The  Germans  and  Russians  employed  a  nearly  similar  apparatus,  but 
with  the  important  difference  that  at  the  principal  focus  of  the  object-glass 
they  inserted  a  plate  of  glass  ruled  with  squares.  These  squares  are  photo- 
graphed upon  the  image  of  the  sun,  and  furnish  a  very  satisfactory  means 
of  determining  the  scale  and  distortion,  if  any,  of  the  image.  The  object- 
glasses  used  by  the  English  and  the  Germans  had  a  focal  length  of  seven 
or  eight  feet.  The  French  employed  object-glasses  with  a  focal  length  of 
some  fourteen  feet,  the  telescope  being  horizontal,  while  the  rays  of  the  sun 
were  reflected  into  it  by  a  plane  mirror ;  instead  of  glass  plates  they  used  the 
old-fashioned  metallic  daguerreotype  plates,  in  order  to  avoid  any  possible 
"  creeping  "  of  the  collodion  film,  which  was  feared  in  the  more  modern  wet- 
plate  process.  The  French  plates  furnish,  however,  no  accurate  orientation 
of  the  picture. 

686.  The  American  Apparatus.  —  The  Americans  used  a  similar  plan, 
with  some  modifications  and  additions.  The  telescope  lenses  employed  were 
five  inches  in  diameter  and  forty  feet  in  focal  length,  so  that  the  image 
directly  formed  upon  the  plate  was  about  4}  inches  in  diameter,  and  needed 
no  enlargement.  The  telescope  was  placed  horizontal  and  in  the  meridian, 
its  exact  direction  being  determinable  by  a  small  transit  instrument  which 
was  mounted  in  such  a  manner  that  it  could  look  into  the  photograph  tele- 
scope, as  into  a  collimator,  when  the  reflector  was  removed.  The  'reflector 
itself  was  a  plane  mirror  of  unsilvered  glass  driven  by  clock-work.  Fig.  193 
shows  the  arrangement  of  the  apparatus.  In  front  of  the  photographic 
plate,  and  close  to  it,  was  supported  a  glass  plate  ruled  with  squares  called 
the  "  reticle  plate,"  and  in  the  narrow  space  between  this  and  the  photograph 


FIG.  193.  —  American  Apparatus  for  Photographing  the  Transit  of  Venus. 


TRANSIT  OF  VENUS  PHOTOGRAPHY. 


421 


FIG.  194.  — Photograph  of  Transit  of  Venus. 


plate  was  suspended  a  plumb-line  of  fine  silver  wire,  the  image  of  which 
appeared  upon  the  plate,  and  gave  the  means  of  determining  the  orientation 

of  the  image  with  extreme 
precision.  If  the  reflec- 
tor were,  and  would  con- 
tinue to  be,  perfectly  plane 
through  the  whole  opera- 
tion, the  method  could 
not  fail  to  give  extremely 
accurate  results ;  but  the 
measurements  and  discus- 
sion of  the  observations 
seem  to  show  that  this 
mirror  was  actually  dis- 
torted to  a  considerable 
extent  by  the  rays  of  the 
sun.  On  the  whole  the 
American  plates  do  not 
appear  to  be  much  more 
trustworthy  than  those 
obtained  by  other  meth- 
ods. Fig.  194  is  a  re- 
duced copy  of  one  of  the  photographs  made  at  Princeton  during  the  transit 
of  1882.  The  black  disc  near  the  middle,  with  a  bright  spot  in  the  centre, 
is  the  image  of  a  metal  disc  cemented  to  the  reticle  to  mark  the  centre  lines 
of  the  reticle  plate;  192  plates  were  taken  during  the  transit,  and  at  some  of 
the  stations  where  the  weather  was  good  the  number  was  much  greater  — 
nearly  300  in  some  cases. 

The  difficulties  to  be  encountered  are  numerous.  Photographic  irradiation, 
or  the  spread  of  the  image  on  the  plate,  slight  distortion  of  the  image  by  the 
lenses  or  mirrors  employed,  irregularities  of  atmospheric  refraction,  uncer- 
tainty as  to  the  precise  scale  of  the  picture,  —  all  these  present  themselves 
in  a  very  formidable  manner.  It  is  obvious  why  this  should  be  so,  when  we 
recall  that  on  a  four -inch  picture  of  the  sun's  disc,  To'Votf  of  an  inch  corre- 
sponds to  about  ^L  of  a  second  of  arc,  and  the  whole  uncertainty  as  to  the 
solar  parallax  does  not  amount  to  as  much  as  that.  An  image  of  the  sun, 
therefore,  in  which  the  position  of  Venus  upon  the  sun's  disc  cannot  be 
determined  accurately  without  an  error  exceeding  T^¥ff  °f  an  inch,  is  of 
very  little  value.  Imperfections  that  would  be  of  no  account  whatever  in 
plates  taken  for  any  other  purpose  make  them  practically  worthless  for  this. 

The  mean  result  of  the  photographic  measures,  using  the  weights 
assigned  by  Newcomb,  gives  a  parallax  of  8".834. 

Newcomb  in  his  "Astronomical  Constants"  combines  the  heliom- 
eter  measures  with  the  photographic,  and  as  a  result  of  all  the 
measures  of  the  position  of  Venus  upon  the  sun's  disc  during  the  transit 
gives  8 ".857  ±0".023;  calling  attention  to  the  fact  that  measures 


422 

of  this   kind   seem  to  be  affected  by  some  constant,  systematic 
error. 

On  the  whole,  the  outcome  of  the  two  transits  of  1874  and  1882 
has  been  to  satisfy  astronomers  generally  that  other  methods  of 
determining  the  sun's  parallax  are  more  to  be  trusted. 

GKAYITATIONAL    METHODS. 

These  are  too  recondite  to  permit  any  full  explanation  here.  We 
can  only  indicate  briefly  the  principles  involved. 

687.  (1)  The  first  of  these  methods  is  by  the  moon's  parallactic 
inequality,  an  irregularity  in  the  moon's  motion  which  has  received 
this  name,'  because  by  means  of  it  the  sun's  parallax  can  be  deter- 
mined.     It  depends  upon  the  fact  that  the  sun's  disturbing  action 
upon  the  moon  differs  sensibly  from  what  it  would  be  if  its  distance, 
instead  of  being  less  than  400  times  that  of  the  moon  from  the  earth, 
were  infinitely  great. 

The  disturbing  action  upon  the  half  of  the  moon's  orbit  which  lies 
nearest  the  sun  is  greater  than  on  the  opposite  half  of  the  orbit.  The 
retarding  action  of  the  tangential  force,  therefore,  during  the  first 
quarter  after  new  moon,  is  perceptibly  greater  than  the  acceleration 
produced  during  the  second  quarter  (Art.  447) ,  so  that  at  the  first  and 
third  quarters  respectively,  the  moon  is  a  little  more  than  2'  behind 
and  ahead  of  the  place  she  would  occupy  if  the  tangential  forces  were 
equal  in  all  four  quadrants  of  the  orbit  —  as  they  would  be  if  the  sun's 
distance  were  infinite.  This  puts  the  moon  about  four  minutes  of  time 
behindhand  at  the  first  quarter,  and  as  much  ahead  at  the  third  ;  and 
if  the  centre  of  the  moon  could  be  observed  within  a  fraction  of  a  second 
of  arc  (as  it  could  if  she  were  a  mere  point  of  light  like  a  star),  the 
observations  would  give  a  very  accurate  determination  of  the  sun's 
distance.  The  irregularities  of  the  moon's  limb,  however,  and  the 
worse  fact,  that  at  the  first  quarter  we  observe  the  western  limb, 
while  at  the  third  quarter  it  is  the  eastern  one  which  alone  is  ob- 
servable, make  the  result  somewhat  uncertain,  though  the  method 
certainly  ranks  high. 

688.  (2)  The  "lunar  equation  of  the  sun's  motion"  is,  it  will  be  remem- 
bered, an  apparent  slight  monthly  displacement  of  the  sun,  amounting  to 
about  6".4,  and  due  to  the  fact  that  both  earth  and  moon  revolve  around  their 
common  centre  of  gravity.     It  is  generally  made  use  of  (Art.  243)  to  deter- 
mine the  mass  of  the  moon  as  compared  with  that  of  the  earth,  using  as  a 


GRAVITATIONAL   METHODS.  423 

datum  the  assumed  known  distance  of  the  sun  ;  but  if  we  consider  the  mass 
of  the  moon  as  known  (determined  by  the  tides,  for  instance),  then  we  can 
find  the  sun's  parallax1  in  terms  of  the  lunar  equation. 

689.  (3)  The  third  method  (by  the  earth's  perturbations  of  Venus 
and  Mars)  is  one  of  the  most  important  of  the  whole  list.  It  depends 
upon  the  principle  that  if  the  mass  of  the  earth,  as  compared  with 
that  of  the  sun,  be  accurately  known,  then  the  distance  of  the  sun 
can  be  found  at  once.  The  reader  will  remember  that  in  Art.  278 
the  mass  of  the  sun  was  found  by  comparing  the  distance  which  the 
earth  falls  towards  the  sun  in  a  second  (as  measured  by  the  curvature 
of  her  orbit)  with  the  force  of  gravity  at  the  earth's  surface ;  and 
in  the  calculation  the  sun's  distance  enters  as  a  necessary  datum. 
Now,  if  we  know  independently  the  sun's  mass  compared  with  the 
earth's,  the  distance  becomes  the  only  unknown  quantity,  and  can 
be  found  from  the  other  data. 

In  the  same  way  as  in  Art.  536  we  have 

** 

)  =  — 

in  which  S  and  E  are  the  masses  of  the  sun  and  earth,  G  is  the  "  constant 
of  gravitation,"  D  is  the  mean  distance  of  the  earth  from  the  sun,  and  T  the 
number  of  seconds  in  a  year.  Also  we  have  for  the  force  of  gravity  at  the 
earth's  surface, 


in  which  r  is  the  earth's  radius. 

Dividing  the  preceding  equation  by  this,  we  get 

S  +  E  _  47r2/^3 
E      ~ 


whence 


O 

If  we  put  —  =  M,  this  becomes 
hi 


gT*r\ 


1  Putting  L  for  the  maximum  value  of  the  lunar  equation  (about  6".  4  of  arc), 
P  for  the  sun's  parallax,  and  R  and  r  for  the  distance  of  the  inoon  and  the  semi- 
diameter  of  the  earth  respectively,  we  have  the  equation 


but  the  numbers  used  are  only  approximate. 


424  DETERMINATION    OF    THE    SUN'S    DISTANCE. 

In  this  equation  everything  in  the  second  term  is  known  when  we  have 
once  found  M,  or  the  ratio  between  the  masses  of  the  sun  and  earth ;  g  is 
found  by  pendulum  observations  on  the  earth,  T  is  the  length  of  the  year 
in  seconds,  and  r  is  the  earth's  radius. 

Now,  the  disturbing  force  of  the  earth  upon  its  next  neighbors, 
Mars  and  Venus,  depends  directly  upon  its  mass  as  compared  with 
the  sun's  mass,  and  the  ratio  of  the  masses  can  be  determined  when 
the  perturbations  have  been  accurately  ascertained;  though  the  cal- 
culation is,  of  course,  anything  but  simple.  But  the  great  beauty  of 
the  method  lies  in  this,  that  as  time  goes  on,  and  the  effect  of  the 
earth  upon  the  revolution  of  the  nodes  and  apsides  of  the  neighbor- 
ing orbits  accumulates,  the  determination  of  the  earth's  mass  in  terms 
of  the  sun's  becomes  continually  and  cumulatively  more  precise.  Even 
at  present  the  method  ranks  high  for  accuracy,  —  so  high  that  Lever- 
rier,  who  first  developed  it,  would  have  nothing  to  do  with  the  transit 
of  Venus  observations  in  1874,  declaring  that  all  such  old-fashioned 
ways  of  getting  at  the  sun's  parallax  were  relatively  of  no  value. 
The  method  is  probably  the  "method  of  the  future,"  and  in  time  will 
supersede  all  the  others,  —  unless  indeed  it  should  appear  that  bodies 
at  present  unknown  are  interfering  with  the  movements  of  our 
neighboring  planets,  or  unless  it  should  turn  out  that  the  law  of 
gravitation  is  not  quite  so  simple  as  it  is  now  supposed  to  be. 

From  this  method  Newcomb  deduces  a  parallax  of  8".759  ±  O^.OIO.  The 
smallness  of  the  value  thus  obtained  is  almost  as  perplexing  at  present  as 
the  magnitude  of  that  derived  from  the  measures  of  the  transits  of  Venus. 


THE    PHYSICAL    METHOD. 

690,  The  physical,  or  "photo-tachy-metrical"  method,  as  it  has 
been  dubbed,  depends  upon  the  fundamental  assumption  that  light 
travels  in  interplanetary  space  with  the  same  velocity  as  in  vacuo. 
This  is  certainly  very  nearly  true,  and  probably  exactly  so,  though 
we  cannot  yet  prove  it. 

By  the  recent  experiments  of  Michelson  and  Newcomb  in  this 
country,  following  the  general  method  of  Foucault,  the  velocity  of 
light  has  been  ascertained  with  very  great  precision  and  may  be 
taken  as  299860  kilometres,  or  186330  miles,  with  a  probable  error 
which,  cannot  well  be  as  great  as  twenty-five  miles  either  way. 

691,  Sun's  Distance  from  the  Equation  of  Light.  —  (1)  "  The 

equation  of  light "  is  the  time  occupied  by  light  in  travelling  between 


SUN'S   DISTANCE   FROM   CONSTANT    OF   ABERRATION.       425 

the  sun  and  earth,  and  is  determined  by  observation  of  the  eclipses 
of  Jupiter's  satellites  (Art.  629).  By  simply  multiplying  the  veloc- 
ity of  light  by  this  time  (499s  ±  2s)  we  have  at  once  the  sun's  dis- 
tance ;  and  that  independent  of  all  knowledge  as  to  the  earth's  dimen- 
sions. The  reader  will  remember,  however,  that  the  determination 
of  this  "light-equation"  is  not  yet  so  satisfactory  as  desirable  on 
account  of  the  indefinite  nature  of  the  eclipse  observations  involved. 

692.  From  the  Constant  of  Aberration.1  —  (2)  When  we  know  the 
velocity  of  light  we  can  also  derive  the  sun's  distance  from  the 
"constant  of  aberration,"  and  this  constant,  20".47,  derived  from 
star  observations  (Art.  225),  is  known  with  a  considerably  higher 
percentage  of  accuracy  than  the  light-equation. 

Calling  the  constant  a,  we  have 

tan  a  =  —  » 

where  U  is  the  velocity  of  the  earth  in  its  orbit,  and  V  the  velocity 
of  light.  Now  U  equals  the  circumference  of  the  earth's  orbit 
divided  by  the  length  of  the  year  ;  i.e., 


T 

hence         tana=—  ,         and         D  =  -* 

On  the  whole  it  seems  likely  at  present  that  the  value  of  the  sun's 
distance  thus  derived  is  the  most  accurate  of  all.  Using  a  =  20".47 
and  V=  186330  miles,  we  have  D  =  92  876000  miles,  and  taking  the 
earth's  equatorial  radius  as  3963.296  miles  (Clarke,  1878),  we  get 
8  ".803  as  the  sun's  equatorial  horizontal  parallax.  But  it  is  to  be 
noted  that  the  parallax  appears  only  as  a  secondary  result.  The 
method  gives  directly  the  distance  of  the  sun,  without  demanding 
any  knowledge  of  the  earth's  dimensions.  The  velocity  of  light  fur- 
nishes the  scale  of  miles. 

693.  The  reader  will  notice  that  the  geometrical  methods  give  the 
parallax  of  the  sun  directly,  apart  from  all  hypothesis  or  assumption, 
except  as  to  the  accuracy  of  the  observations  themselves,  and  of 
their  necessary  corrections  for  refraction,  etc.  :  the  gravitational 
methods,  on  the  other  hand,  assume  the  exactness  of  the  law  of 

1  See  note  on  page  427. 


426  DETERMINATION   OF   THE   SUN'S   DISTANCE. 

gravitation ;  and  the  physical  method  assumes  that  light  travels  in 
space  with  the  same  velocity  as  in  our  terrestrial  experiments  after 
allowing  for  the  known  retardation  due  to  the  refracting  power  of 
the  air.  The  near  accordance  of  the  results  obtained  by  the  different 
methods  shows  that  these  assumptions  must  be  very  nearly  correct, 
if  not  absolutely  so. 

NOTE  TO  ARTICLE  676*. 

During  the  winter  of  1901-2  the  planet  Eros  came  quite  near  the  earth,  and 
an  extensive  series  of  observations,  both  micrornetric  and  photographic,  was 
made  upon  it  at  numerous  cooperating  observatories  for  the  purpose  of  deter- 
mining the  solar  parallax. 

It  will  be  some  time  (perhaps  several  years)  before  the  immense  mass  of  data 
can  be  thoroughly  discussed  and  the  final  result  determined  ;  but  already  partial 
reductions  of  the  observations  made  at  a  few  of  the  stations  show  that  the  cor- 
rection to  the  assumed  parallax  (8". 80)  will  be  extremely  small ;  nor  is  it  even 
certain  whether  the  correction  will  be  in  the  direction  of  increase  or  decrease. 

NOTE. 

Newcomb,  in  his  "Astronomical  Constants"  (1896),  adopts  8".797 
±  0".007  as  the  value  of  the  solar  parallax  to  be  used  in  the  plane- 
tary tables. 

He  also  gives  the  following  as  the  results  derived  by  the  various  methods 
after  making  allowance  for  probable  systematic  errors,  and  assigns  to  each  result 
the  weight  indicated  by  the  number  that  follows  it. 

Motion  of  the  Node  of  Venus 8".7GS,  10 

GilVs  Observations  of  Mars  (1877) 8.780,     1 

Pulkowa  Constant  of  Aberration  (20".492)  .     .     .     .  8.793,40 

.     Contact  Observations  of  Transit  of  Venus     ....  8  .794,     3 

Heliometer  Observations  of  Victoria  and  Sappho    .     .  8  .799,     5 

Parallactic  Inequality  of  the  Moon 8  .794,  10 

Miscellaneous  Determinations  of  Aberration  (20".463)  8  .806,  10 

Lunar  Inequality  of  the  Earth 8  .818,     1 

Measures  of  Venus  in  Transit 8  .857,     1 

While  many  would  question  the  overwhelming  weight  assigned  to  the 
Pulkowa  aberration,  it  makes  very  little  difference  in  the  result. 

Harkness,  in  his  "  Solar  Parallax  and  its  Belated  Constants " 
(1891),  obtained  as  his  final  value  8".809  ±  0".006. 


EXERCISES.  427 

EXERCISES  ON  CHAPTER  XVII. 

^  1.   Which  of  the  methods  of  determining  the  distance  of  the  sun  does 
not  presuppose  a  knowledge  of  the  dimensions  of  the  earth  ? 

2.  Why  cannot  the  transits  of  Mercury  be  utilized  for  determining  the 
solar  parallax  as  well  as  the  transits  of  Venus  ? 

'  3.    Can  observations  of  Jupiter  or  any  of  the  remoter  planets  be  usefully 
employed  in  determining  the  sun's  distance? 

4.  How  much  error  in  the  distance  of  the  sun  follows  from  an  error  of 
0".01  in  the  value  of  the  parallax? 

5.  How  much  error  in  the  distance  of  the  sun  follows  from  an  error  of 
one  per  cent  in  the  value  of  the  ratio  between  the  masses  of  the  earth  and 
sun  as  determined  from  planetary  perturbations  ? 

6.  Could   the   parallax  of   the   sun    be  determined  within  ten  per  cent 
without  the  use  of  the  telescope? 

NOTE  TO  ARTS.  '174  AND  692. 

The  spectrographic  measurements  of  the  radial  velocity  of  stars  with  respect 
to  the  earth  (Art.  802*)  also  furnish  an  independent  determination  of  her  orbi- 
tal velocity  in  terms  of  the  velocity  of  light,  and  so  of  our  distance  from  the  sun. 

In  the  spectra  of  all  stars  near  the  ecliptic,  the  dark  lines  are  found  to  shift 
regularly  backward  and  forward  during  the  year  by  an  amount  which  indicates 
a  range  of  about  37  miles  a  second  in  their  radial  velocity  relative  to  the  earth, 
or  an  orbital  velocity  of  the  earth  equal  to  about  18.5  miles  a  second. 

The  first  application  of  this  method  has  been  made  by  Klistner  in  1905.  From 
a  series  of  spectroscopic  determinations  of  the  changes  in  the  radial  velocity  of 
Arcturus  in  1904-5  he  deduces  a  solar  parallax  of  8". 84.  As  yet,  however, 
the  method  cannot  compete  in  precision  with  the  determination  by  means  of 
aberration ;  the  result  is  rather  to  be  regarded  as  a  confirmation  of  Doppler's 
principle. 


428  COMETS. 


CHAPTER    XVIII. 


COMETS:    THEIR  NUMBER,  MOTIONS,  AND  ORBITS.  -  THEIR 

STITUENT    PARTS    AND    APPEARANCE.  -  THEIR    SPECTRA.  - 
THEIR   PHYSICAL   CONSTITUTION,    AND   ORIGIN. 

694.  FROM  time  to  time  bodies  of  a  very  different  character  from 
the  planets  make  their  appearance  in  the  heavens,  remain  visible 
for  some  weeks  or  months,  move  over  a  longer  or  shorter  path  among 
the  stars,  and  then  vanish.     These  are  the  COMETS,  or  "  hairy  stars" 
as  the  word  means,  since  the  appearance  of  such  as  are  bright 
enough  to  be  visible  to  the  naked  eye  is  that  of  a  star  surrounded 
by  a  hazy  cloud,  and  usually  carrying  with  it  a  streaming  trail  of 
light.     Some  of  them  have  been  magnificent  objects,  —  the  nucleus, 
or  central  star,  as  brilliant  as  Venus  and  visible  even  by  day,  while 
the  cloudy  head  was  nearly  as  large  as  the  sun  itself,  and  the  tail 
extended  from  the  horizon  to  the  zenith,  —  a  train  of  shining  sub- 
stance long  enough  to  reach  from  the  earth  to  the  sun.     The  major- 
ity of  comets,  however,  are  faint,  and  visible  only  with  a  telescope. 

695.  Superstitions.  —  In  ancient  times  these  bodies  were  regarded 
with  great  alarm  and  aversion,  being  considered  from  the  astrological  point 
of  view  as  always  ominous  of  evil.      Their  appearance  was  supposed  to 
presage  war,  or  pestilence,  or  the  death  of  princes.     These  notions  have 
survived  until  very  recent  times  with  more  or  less  vigor,  but,  it  is  hardly 
necessary  to  say,  without  the  least  reason.     The  most  careful  research  fails 
to  show  any  effect  upon  the  earth  produced  by  a  comet,  even  of  the  largest 
size.     There  is  no  observable  change  of  temperature  or  of  any  meteoro- 
logical condition,  nor  any  effect  upon  vegetable  or  animal  life. 

696.  Number  of  Comets.  —  Thus    far   we    have    on    our    lists 
nearly  700,  including  the  different  returns  of  the  periodic  comets. 
About  400  were  recorded  previous  to  1600,  before  the  invention  of 
the  telescope,  and  must,  of  course,  have  been  fairly  conspicuous. 
Since  that  time  the  number  annually  observed  has  increased  very 
greatly,  for  only  a  few  of  these  bodies,  perhaps  one  in  five,  are 
visible  without  telescopic  aid.     Their  total  number  must  be  enor- 
mous.    Not  unfrequently  from  five  to  eight  are  discovered  in  a 


DESIGNATION   OF   COMETS.  429 

single  year,  and  there  is  seldom  a  day  when  one  or  more  is  not  in 
sight. 

While  telescopic  comets,  however,  are  thus  numerous,  brilliant 
ones  are  comparatively  rare.  Between  1500  and  1800  there  were, 
according  to  Newcomb,  79  visible  to  the  naked  eye,  or  about  one 
in  three  and  three-fourths  years.  Humboldt  enumerates  43  within 
the  same  period  as  conspicuous;  during  the  first  half  of  the  present 
century  there  were  9  such,  and  since  1850  there  have  been  14.  Since, 
and  including,  1880  we  have  had  9,  —  a  remarkable  number  for  so 
short  a  time,  —  and  two  of  them,  the  principal  comet  of  1881  and 
the  great  comet  of  1882,  were  unusually  fine  ones.  In  August,  1881, 
for  a  little  time  two  comets  were  conspicuously  visible  to  the  naked 
eye  at  once  and  near  together  in  the  sky,  a  thing  almost  if  not  quite 
unprecedented. 

697.  Designation   of  Comets. — The  more  remarkable  ones  gen- 
erally bear  the  name  of  their  discoverer,  or  of  some  astronomer  who 
made  important  investigations  relating  to  them,  —  as  for  instance, 
Halley's,  Encke's,  and  Donati's  comets.     They  are  also  designated 
by  the  year  of  discovery,  with  a  Roman  number  indicating  the  order 
of  perihelion  passage.     A  third  method  of  designation  is  by  year  and 
letter,  the  letters  denoting  the  order  in  which  the  comets  of  a  given 
year  were  discovered.     Thus  Donati's  comet  was  both  comet  f  and 
comet  VI,   1858.     Comet  I  is,   however,   not  necessarily  comet    &, 
though  it  usually  is  so.     In  some  cases  the  comet  bears  the  name 
of   two  discoverers.      Thus   the  Pons-Brooks  comet   of  1883   is   a 
comet  which  was  discovered  by  Pons  in  1812,  and  at  its  return  in 
1883  was  discovered  by  Brooks. 

698.  The  Discovery  of  Comets.  —  As  a  rule  these  bodies  are  first  seen 
by  comet-hunters,  who  make  a  business  of  searching  for  them.     For  such 
purposes  they  are  usually  provided  with  a  telescope  known  as  a  "  comet- 
seeker,"  with  an  eye-piece  of  low  power,  and  a  large  field  of  view.    When  first 
seen,  a  comet  is  usually  a  mere  roundish  patch  of  faintly  luminous  cloud, 
which,  if  really  a  comet,  will  reveal  its  true  character  within  an  hour  or  two 
by  its  motion.     Some  observers  have  found  a  great  number  of  these  bodies. 
Messier  discovered  13  between  1760  and  1798,  and  Pons  27  between  1800 
and  1827.     In  the  U.  S.,  Brooks,  Barnard,  and  Swift  have  been  especially 
successful. 

699.  Duration  of  Visibility,  and  Brightness.  —  The  time  during 
which,  they  are  visible  differs  very  much.     The  great  comet  of  1811 


430  COMETS. 

was  observed  for  seventeen  months.  Comet  1889  I  was  followed  at 
the  Lick  Observatory  for  more  than  two  years ;  and  probably  with 
our  present  instruments  it  will  be  possible  to  prolong  observations 
far  beyond  limits  formerly  possible.  When  a  comet  is  not  discovered 
until  it  is  receding  from  the  sun  it  is  sometimes  observable  only  for 
a  few  weeks  or  days. 

As  to  their  brightness  they  also  differ  widely.  The  great  majority 
can  be  seen  only  with  a  telescope,  although  a  considerable  number 
reach  the  limit  of  naked-eye  vision  at  that  part  of  their  orbit  where 
they  are  most  favorably  situated.  A  few,  as  has  been  said  above, 
become  conspicuous  ;  and  a  very  few,  perhaps  four  or  five  in  a  century, 
are  so  brilliant  that  they  can  be  seen  by  the  naked  eye  in  full  sun- 
light, as  was  the  case  with  the  great  comets  of  1843  and  1882. 

700.  Their  Orhits.  —  The  ideas  of  the  ancients  as  to  the  motions  of 
these  bodies  were  very  vague.  Aristotle  and  his  school  believed  them  to  be 
nothing  but  earthly  exhalations  inflamed  in  the  upper  regions  of  the  air, 
and  therefore  meteorological  phenomena  rather  than  astronomical.  Ptolemy 
accordingly  omits  all  notice  of  them  in  the  Almagest. 

Tycho  Brahe  was  the  first  to  show  that  they  are  more  distant  than  the 
moon  by  comparing  observations  of  the  comet  of  1577  made  in  different 
parts  of  Europe.  Its  position  among  the  stars  at  any  moment,  as  seen  from 
his  observatory  at  Uranienburg,  was  sensibly  the  same  as  that  observed  at 
Prague,  more  than  400  miles  to  the  south.  It  followed  that  its  distance 
must  be  much  greater  than  that  of  the  moon,  and  that  its  real  orbit  must  be 
of  enormous  size,  cutting  through  interplanetary  space  in  a  manner  abso- 
lutely incompatible  with  the  old  doctrine  of  the  crystalline  spheres.  He 
supposed  the  path  to  be  circular,  however,  as  befitted  the  motion  of  a 
celestial  body. 

Kepler  supposed  that  comets  moved  in  straight  lines ;  and  he  seems  to 
have  been  half  disposed  to  consider  them  as  living  creatures,  travelling 
through  space  with  will  and  purpose,  "like  fishes  in  the  sea." 

Hevelius  first,  nearly  a  hundred  years  later,  suggested  that  the  orbits  are 
probably  parabolas,  and  his  pupil  Doerfel  proved  this  to  be  the  case  in  1681 
for  the  comet  of  that  year.  The  theory  of  gravitation  had  now  appeared, 
and  Newton  soon  worked  out  and  published  a  method  by  which  the  ele- 
ments of  a  comet's  orbit  can  be  determined  from  the  observations.  Imme- 
diately afterwards  Halley,  using  this  method  and  computing  the  parabolic 
orbits  of  all  the  comets  for  which  he  could  find  the  needed  observations, 
ascertained  that  a  series  of  brilliant  comets  having  nearly  the  same  orbit 
had  appeared  at  intervals  of  about  seventy-five  years.  He  concluded  that 
these  were  different  appearances  of  one  and  the  same  comet,  the  orbit  not 
being  really  parabolic  but  elliptical,  and  he  predicted  its  return,  which 
actually  occurred  in  1759  —  the  first  of  "  periodic  comets." 


DETERMINATION  OF  A  COMET'S   ORBIT.  431 

701.  Determination  of  a  Comet's  Orbit.  —  Strictly  speaking,  the 
orbit  of  a  comet  being  always  a  conic  section,  like  that  of  a  planet, 
requires  only  three  perfect  observations  for  its  determination ;  but  it 
seldom  happens  that  the  observations1  can  be  made  so  accurately 
as  to  enable  us  to  distinguish  an  orbit  truly  parabolic  from  one 
slightly  hyperbolic,  or  from  an  ellipse  of  long  period.  The  plane 
of  the  orbit  and  its  perihelion  distance,  can  be  made  out  with  reason- 


FIG.  195. 
The  Close  Coincidence  of  Different  Species  of  Cometary  Orbits  within  the  Earth's  Orbit. 

able  accuracy  from  such  observations  as  are  practically  obtainable, 
but  the  eccentricity,  and  the  major  axis  with  its  corresponding  period, 
can  seldom  be  determined  with  much  precision  from  the  data  obtained 
at  a  single  appearance  of  a  comet  unless  its  orbit  is  small. 

The   reason  is   that  a  comet  is  visible  only  in  that  very  small 

1  Observations  for  the  determination  of  a  comet's  place  are  usually  made  with 
an  equatorial,  by  measuring  the  apparent  distance  between  the  comet  and  some 
neighboring  "  comparison  star "  with  some  form  of  micrometer,  as  indicated  in 
Art.  129.  If  the  star's  place  is  not  already  accurately  known,  it  is  afterwards 
specially  observed  with  the  meridian  circle  of  some  standard  observatory ;  this 
observation  of  comparison  stars  forms  quite  an  item  in  the  regular  work  of  such 
an  institution. 


432  COMETS. 

portion  of  its  orbit  which  lies  near  the  earth  and  sun,  and,  as  the 
figure  shows  (Fig.  195),  in  this  portion  of  the  orbit,  the  long  ellipse, 
the  parabola,  and  the  hyperbola  almost  coincide.  Moreover,  from  the 
diffuse  nature  of  a  comet  it  is  not  possible  to  observe  it  with  the 
same  accuracy  as  a  planet. 

Comets  which  really  move  in  parabolas  or  hyperbolas  visit  the  sun 
but  once,  and  then  recede,  never  to  return  ;  while  those  that  move  in 
ellipses  return  in  regular  periods,  unless  disturbed. 

It  will  be  understood,  that  in  a  catalogue  of  comets'  orbits,  those 
which  are  indicated  as  parabolic  are  not  strictly  so.  All  that  can  be 
said  is  that  during  the  time  while  the  comet  was  visible,  its  position 
did  not  deviate  from  the  parabola  given  by  an  amount  sensible  to 
observation.  The  chances  are  infinity  to  one  against  a  comet's 
moving  exactly  in  a  parabola,  since  the  least  retardation  of  its 
velocity  would  render  the  orbit  elliptical,  and  the  least  acceleration, 
hyperbolic,  according  to  the  principles  explained  in  Article  430. 

702.  Relative  Numbers  of  Parabolic,  Elliptical,  and  Hyperbolic 
Comets.  —  The  orbits  of  over  400  comets  have  been  thus  far  com- 
puted. Of  this  number  over  300  are  sensibly  parabolic,  and  about 
a  dozen  have  had  orbits  which  were  hyperbolic  according  to  some 
computation  or  other ;  in  no  single  case,  however,  is  the  hyperbolic 
character  certain,  though  in  two  it  is  very  probable.  There  are  also 
a  number  of  comets  which,  according  to  the  best  computations, 
appear  to  have  orbits  really  elliptical,  but  with  periods  so  long  that 
their  elliptical  character  cannot  be  positively  asserted.  About 
ninety  have  orbits  which  are  certainly  and  distinctly  oval;  and 
sixty-five  of  these  have  periods  which  are  less  than  one  hundred 
years.  Eighteen  of  these  periodic  comets  have  already  been  actually 
observed  at  more  than  one  return  (May,  1908). 

As  to  the  rest  of  the  sixty-five,  some  of  them  are  expected  to  return  again 
within  a  few  years,  and  some  of  them  have  been  lost,  —  either  in  the  same 
way  as  the  comet  of  Biela,  of  which  we  shall  soon  speak,  or  by  having  their 
orbits  so  changed  by  perturbations  that  they  no  longer  come  near  enough 
to  the  earth  to  be  observed.  See  Appendix,  Table  III. 

There  are  three  comets  with  computed  periods  ranging  between  seventy 
and  eighty  years,  whose  returns  are  looked  for  within  the  next  forty  years. 
There  is  also  one  comet  with  a  period  of  thirty-three  years  which  was  due 
to  return  in  1899,  but  failed  to  appear.  It  is  known  as  Tempers  comet, 
an  inconspicuous  body,  but  of  great  interest  from  its  connection  with  the 
"  Leonid  Meteor-Swarm." 


RECOGNITION   OF  ELLIPTIC   COMETS. 


433 


.  703.  Fig.  196  shows  the  orbits  of  several  of  the  comets  of  short 
period, — from  three  to  eight  years.  (It  would  cause  confusion  to 
insert  all  of  them.)  It  will  be  seen  that  in  every  case  the  comet's 
orbit  comes  very  near  to  the  orbit  of  Jupiter,  and  when  the  orbit 
crosses  that  of  Jupiter,  one 
of  the  nodes  is  always  near 
the  place  of  apparent  in- 
tersection (the  node  being 
marked  on  the  comet's  orbit 
by  a  short  cross-line).  If 
Jupiter  were  at  that  point 
of  its  orbit  at  the  time  when 
the  comet  was  passing,  the 
two  bodies  would  really  be 
very  near  to  each  other. 
The  fact,  as  we  shall  see, 
is  a  very  significant  one, 
pointing  to  a  connection  be- 
tween these  bodies  and  the 
planet.  It  is  true  for  all 
the  comets,  whose  periods 
are  less  than  eight  years  — 
for  those  not  inserted  in 

the  diagram  as  well  as  those  that  are.  The  orbits  of  the  seventy-five- 
year  comets  are  similarly  related  to  the  orbit  of  Neptune,  and  the 
thirty-three-year  comet  passes  very  close  to  the  orbit  of  Uranus. 


FIG.  196.  —  Orbits  of  Short-period  Comets. 


704.  Recognition  of  Elliptic  Comets.  —  Modern  observations  are 
so  much  more  accurate  than  those  made  two  centuries  ago  that  it  is 
now  sometimes  possible  to  determine  the  eccentricity  and  period  of 
an  elliptic  comet  by  means  of  the  observations  made  at  a  single 
appearance.  Still,  as  a  general  rule,  it  is  not  safe  to  pronounce  upon 
the  ellipticit}'  of  a  comet's  orbit  until  it  has  been  observed  at  least 
twice,  nor  always  then.  A  comet  possesses  no  "personal  identity," 
so  to  speak,  by  which  it  can  be  recognized  merely  "by  looking  at  it,  — 
no  personal  peculiarities  like  those  of  the  planets  Jupiter  and  Saturn. 
It  is  identifiable  only  by  its  path. 

When  the  approximate  parabolic  elements  of  a  new  comet's  orbit  have 
been  computed,  we  examine  the  catalogue  of  preceding  comets  to  see  if  we 
can  find  others  which  resemble  it ;  that  is,  which  have  nearly  the  same  incfo 


434  COMETS. 

nation  and  longitude  of  the  node  with  the  same  perihelion  distance  and 
helion  longitude.  If  so,  it  is  probable  that  we  have  to  do  with  the  same  comet 
in  both  cases.  But  it  is  not  certain,  and  investigations,  often  very  long  and 
intricate,  must  be  made  to  see  whether  an  elliptical  orbit  of  the  necessary 
period  can  be  reconciled  with  the  observations,  after  taking  into  account 
the  perturbations  produced  by  planetary  action.  These  perturbations  are 
extremely  troublesome  to  compute,  and  are  often  very  great,  since  the  comets 
not  unfrequently  pass  near  to  the  larger  planets.  In  some  such  cases  the 
orbit  is  completely  altered.  Even  if  the  result  of  this  investigation  appears 
to  show  that  the  comets  are  probably  identical,  we  are  not  yet  absolutely 
safe  in  the  conclusion,  for  we  have  what  are  known  as  — 

705.  Cometary  Groups.  —  These   are   groups   of  comets   which 
pursue  nearly  the  same  orbits,  following  along  one  after  another  at  a 
greater  or  smaller  interval,  as  if  they  had  once  been  united,  or  had  come 
from  some  common  source.     The  existence  of  such  groups  was  first 
pointed  out  by  Hoek  of  Utrecht  in  1865.     The  most  remarkable 
group  of  this  sort  is  the  one  composed  of  the  great  comets  of  1668, 
1843,  1880,  and  1882,  and  there  is  some  reason  to  suspect  that  the 
little  comet  visible  on  the  picture  of  the  corona  of  the  Egyptian 
eclipse  (Art.  328)  also  belongs  to  it.     The  bodies  of  this  group  have 
orbits  very  peculiar  in  their  extremely  small  perihelion  distance 
(they  actually  go  within  half  a  million  miles  of  the  sun's  surface), 
and  yet,  although  their  elements  are  almost  identical,  they  cannot 
possibly  all  be  different  appearances  of  one  and  the  same  comet. 

So  far  as  regards  the  comets  of  1668  and  1843,  considered  alone,  there 
is  nothing  absolutely  forbidding  the  idea  of  their  identity :  perturbations 
might  account  for  the  differences  between  their  two  orbits.  But  the  comets 
of  1880  and  1882  cannot  possibly  be  one  and  the  same ;  they  were  both 
observed  for  a  considerable  time  and  accurately,  and  the  observations  of 
both  are  absolutely  inconsistent  with  a  period  of  two  years  or  anything  like 
it.  In  fact,  for  the  comet  of  1882  all  of  the  different  computers  found 
periods  ranging  between  600  and  900  years. 

There  are  about  half  a  dozen  other  such  comet-groups  now  known. 

706.  Perihelion  Distances.  —  These  vary  greatly.    Twelve  comets 
have  a  perihelion  distance  less  than  five  millions  of  miles 1 ;  about 
seventy-four  per  cent  of  all  that  have  been  observed  lie  within  the 
earth's  orbit ;   about  twenty-four  per  cent  lie  outsid,e,  but  within 
twice  the  earth's  distance  from  the  sun ;  and  eleven  comets  have 
been  observed  with  a  perihelion  distance  exceeding  that  limit. 

1  According  to  Galle's  tables.    Other  authorities  give  slightly  different  numbers. 


ORBIT   PLANES.  435 

A  single  one,  the  comet  of  1729,  had  a  perihelion  distance  exceeding  four 
astronomical  units,  —  as  great  as  the  mean  distance  of  the  remoter  asteroids. 
It  must  have  been  an  enormous  comet  to  be  visible  from  such  a  distance. 
One  computer  made  its  orbit  slightly  hyperbolic :  others  did  not. 

Obviously,  however,  the  distribution  of  comets  as  determined  by  observa- 
tion, depends  not  merely  on  the  existence  of  the  comets  themselves,  but  upon 
their  visibility  from  the  earth.  Those  comets  which  approach  near  the  orbit 
of  the  earth  have  the  best  chance  of  being  seen,  because  their  conspicuous- 
ness  increases  as  they  approach  us,  so  that  we  must  not  lay  too  much  stress 
on  the  apparent  crowding  of  the  perihelia  within  the  earth's  orbit. 

The  perihelia  are  not  distributed  equally  in  all  directions  from  the 
sun,  but  more  than  sixty  per  cent  are  within  45°  of  what  is  called 
"  the  sun's  way"  ;  i.e.,  the  line  in  space  along  which  the  sun  is  travel- 
ling, carrying  with  it  its  attendant  S3rstems. 

707.  Orbit  Planes.  — The  inclinations  of  the  comets'  orbits  range 
all  the  way  from  0°  to  90°.     The  ascending  nodes  are  distributed  all 
around  the  ecliptic,  with  a  decided  tendency,  however,  to  cluster  in 
two  regions  having  a  longitude  of  about  80°  and  270°. 

708.  Direction  of  Motion. — With  the  two  exceptions  of  Halley's 
comet,  and  the  comet  of  the  Leonid  meteors  (Art.  786) ,  the  elliptical 
comets  which  have  periods  less  than  one  hundred  years  all  move  in 
the  direction  of  the  planets.     Of  the  other  comets,  a  few  more 
move  retrograde  than  direct,  but  there  is  no  decided  preponderance 
either  way. 

709.  It  is  hardly  necessary  to  point  out  that  the  fact  that  the 
comets  move  for  the  most  part  in  parabolas,  and  that  the  planes  of 
their  orbits  have  no  evident  relation  to  the  plane  of  the  planetary 
motions,  tends  to  indicate  (though  it  falls  short  of  demonstrating) 
that   they   do   not  in   any  proper  sense   belong   to   the   solar  system 
itself,  but  are   merely  visitors  from   interstellar  space.  •  They   come 
towards  the  sun  with  almost  precisely  the  velocity  they  would  have 
if  they  had  simply  dropped  towards  it  from  an  infinite  distance,  and 
they  leave  it  with  a  velocity  which,  if  no  force  but  the  sun's  attrac- 
tion  operates   upon    them,  will   carry   them    back   to    an   unlimited 
distance,  or  until  they  encounter  the  attraction  of  some  other  sun. 
With  one  remarkable  exception,  their  motions  appear  to  be  just  what 
might   be    expected  of   ponderable  masses  moving   in   empty  space 
under  the  law  of  gravitation. 


436  COMETS. 

710,  Acceleration  of  Encke's  Comet.  —  The  one  exception  referred 
to  is  in  the  case  of  Encke's  comet  which,  since  its  first  discovery  in 
the  last  century  (it  was  not,  however,  discovered  to  be  a  periodic 
comet  until  1819),  has  been  continually  quickening  its  speed  and 
shortening  its  period  at  the  rate  of  about  two  hours  and  a  half  in 
each  revolution ;  as  if  it  were  under  the  action  of  some  resistance 
to  its  motion.  No  perturbation  by  any  known  body  will  account  for 
such  an  acceleration,  and  thus  far  no  reasonable  explanation  has 
been  suggested  as  even  possible,  except  that  something  encountered 
in  its  motion  through  interplanetary  space  retards  the  comet  just 
as  air  retards  a  rifle-bullet.  At  first  sight  it  seems  almost  paradoxi- 
cal that  a  resistance  should  accelerate  a  comet's  speed ;  but  referring 
to  Article  429  we  see  that  since  the  semi-major  axis  of  a  comet's 
orbit  is  given  by  the  equation 

U* 


2\U*-1 

any  diminution  of  V  will  also  diminish  a ;  and  it  can  be  shown  that 
this  reduction  in  the  size  of  the  orbit  will  be  followed  by  an  increase 
of  velocity  above  that  which  the  body  had  in  the  larger  orbit.  It 
gains  more  speed  by  thus  falling  into  a  smaller  orbit  nearer  to  the 
sun  than  it  loses  by  the  direct  effect  of  the  resistance. 

711.  Another  action  of  such  a  retarding  force  is  to  diminish  the  eccen- 
tricity of  the  body's  orbit,  making  it  more  nearly  circular.  If  the  action  were 
to  go  on  without  intermission,  the  result  would  be  a  spiral  path  winding  in- 
ward towards  the  sun,  upon  which  the  comet  would  ultimately  fall.  For  many 
years  the  behavior  of  Encke's  comet  was  quoted  as  an  absolute  demonstra- 
tion of  the  existence  of  the  "luminiferous  ether."  Since,  however,  no  other 
comets  show  any  such  action  (unless  perhaps  Winnecke's *  comet  —  No.  5  in 
the  table  in  the  Appendix),  and  moreover,  since  according  to  the  investi- 
gations of  Von  Asten  and  Backlund  the  acceleration  of  Encke's  comet  itself 
seems  suddenly  to  have  diminished  by  nearly  one-half  in  1868,  there  remains 
much  doubt  as  to  the  theory  of  a  resisting  medium.  It  looks  rather  more 
probable  that  this  acceleration  is  due  to  something  else  than  the  luminiferous 
ether  —  perhaps  to  some  regularly  recurring  encounter  of  the  comet  with  a 
cloud  of  meteoric  matter.  The  fact  that  the  planets  show  no  such  effect  is 
not  surprising,  since,  as  we  shall  see,  they  are  enormously  more  dense  than 
any  comet,  so  that  the  resistance  that  would  bring  a  comet  to  rest  within  a 

1  Oppolzer,  in  1880,  found  that  according  to  his  computations  Winnecke's 
comet  was  accelerated  precisely  in  the  same  way  as  Encke's,  but  by  less  than  half 
the  amount.  His  result,  however,  is  not  confirmed  by  the  later  work  of  Hardtl, 
who  finds  no  acceleration  at  all. 


PHYSICAL  CHARACTERISTICS   OF   COMETS.  437 

single  year  would  not  sensibly  affect  a  body  like  our  earth  in  centuries.  The 
*'  resisting  medium,"  if  it  exists  at  all,  must  have  much  less  retarding  power 
than  the  residual  gas  in  one  of  Crookes's  best  vacuum  tubes. 

712.  Physical  Characteristics   of  Comets.  —  The   orbits  of  these 
bodies  are  now  thoroughly  understood,  and  their  motions  are  calcu- 
lable with  as  much  accuracy  as  the  nature  of  the  observations  permit : 
but  we  find  in  their  phj'sical  constitution  some  of  the  most  perplex 
ing  and  baffling  problems  in  the  whole  range  of  astronomy,  —  appar- 
ent paradoxes  which  as  yet  have  received  no  satisfactory  explanation. 
While  comets  are   evidently  subject   to   gravitational  attraction,  as 
shown  by  their  orbits,  they  also  exhibit  evidence  of  being  acted  upon 
by  powerful  repulsive  forces  emanating  from  the  sun.     While  they 
shine,  in  part  at  least,  by  reflected  light,  they  are  also  certainly  self- 
luminous,  their  light  being  developed  in  a  way  not  yet  satisfactorily 
explained.      They    are   the   bulkiest  bodies   known,    in   some    cases 
thousands  of  times  larger  than  the  sun  or  stars  ;  but  they  are  "airy 
nothings,"  and  the  smallest  asteroid  probably  rivals  the  largest  of 
them  in  actual  mass. 

713.  Constituent  Parts  of  a  Comet.  —  (a)  The  essential  part  of  a 
comet  —  that  which  is  always  present  and  gives  it  its  name  —  is  the 
coma  or  nebulosity,  a  hazy  cloud  of  faintly  shining  matter,  which 
is  usually  nearly  spherical  or  oval  in  shape,  though  not  always  so. 

(&)  Next  we  have  the  nucleus,  which,  however,  is  not  found  in  all 
comets,  but  commonly  makes  its  appearance  as  the  comet  approaches 
the  sun.  t  It  is  a  bright,  more  or  less  star-like  point  near  the  centre 
of  the  coma,  and  is  the  object  usually  pointed  on  in  determining  the 
comet's  place  by  observation.  In  some  cases  the  nucleus  is  double 
or  even  multiple  ;  that  is,  instead  of  a  single  nucleus  there  may  be 
two  or  more  near  the  centre  of  a  comet.  Perhaps  three  comets  out  of 
four  present  a  nucleus  during  some  portion  of  their  visibility. 

(c)  The  tail  or  train,  is  a  streamer  of  light  which  ordinarily  ac- 
companies a  bright  comet,  and  is  often  found  even  in  connection 
with  a  telescopic  comet.  As  the  comet  approaches  the  sun,  the  tail 
follows  it  much  as  the  smoke  and  steam  from  the  locomotive  trail  after 
it.  But  that  the  tail  does  not  really  consist  of  matter  simply  left 
behind  in  that  way,  is  obvious  from  the  fact  that  as  the  comet  recedes 
from  the  sun,  the  train  precedes  it  instead  of  following.  It  is  always 
directed  away  from  the  sun,  though  its  precise  position  and  form  is  to 
some  extent  determined  by  the  comet's  motion.  There  is  abundant 
evidence  that  it  is  a  material  substance  in  an  exceedingly  tenuous 


438  COMETS. 

condition,  which  in  some  way  is  driven  off  from  the  comet  and  then 
repelled  by  some  solar  action.     (See  also  Art.  736.) 

(d)  Envelopes  and  Jets.  —  In  the  case  of  a  very  brilliant  comet,  its 
head  is  often  veined  by  short  jets  of  light  which  appear  to  be  con- 


FIG.  197.  —  Naked-eye  View  of  Donati's  Comet,  Oct.  4, 1858.    (Bond.) 

tinually  emitted  by  the  nucleus ;  and  sometimes  instead  of  jets  the 
nucleus  throws  off  a  series  of  concentric  envelopes,  like  hollow  shells, 
one  within  the  other.  These  phenomena,  however,  are  not  usually 
observed  in  telescopic  comets  to  any  marked  extent. 


DIMENSIONS    OF    COMETS.  439 

714.  Dimensions  of  Comets.  —  The  volume  of  a  comet  is  often 
enormous  —  sometimes  almost  beyond  conception,  if  the  tail  be  in- 
cluded in  the  estimate  of  bulk. 

As  a  general  rule  the  head  or  coma  of  a  telescopic  comet  is  from 
40000  to  100000  miles  in  diameter.  A  comet  less  than  10000 
miles  in  diameter  is  very  unusual ;  in  fact,  such  a  comet  would  be 
almost  sure  to  escape  observation.  Many,  however,  are  much  larger 
than  100000  miles.  The  head  of  the  comet  of  1811  at  one  time 
measured  nearly  1 200000  miles,  —  more  than  forty  per  cent  larger 
than  the  diameter  of  the  sun  itself.  The  comet  of  1680  had  a  head 
600000  miles  across.  The  head  of  Donati's  comet  of  1858  was 
250000  miles  in  diameter.  Holmes'  comet  of  1892,  remarkable  in 
many  ways  though  not  brilliant,  had  a  diameter  of  over  700000 
miles,  but  no  visible  nucleus  at  that  time.  A  few  weeks  later  it 
looked  like  a  mere  hazy  star. 

715.  Contraction  of  a  Comet's  Head  as  it  approaches  the  Sun.  —  It 

is  a  very  singular  fact  that  the  head  of  a  comet  continually  changes 
its  diameter  as  it  approaches  to  and  recedes  from  the  sun;  and 
what  is  more  singular  yet,  it  usually  contracts  when  it  approaches 
the  sun,  instead  of  expanding,  as  one  would  naturally  expect  it  to 
do  under  the  action  of  the  solar  heat.  No  satisfactory  explanation  is 
known.  Perhaps  the  one  suggested-  by  Sir  John  Herschel  is  as 
plausible  as  any,  —  that  the  change  is  optical  rather  than  real ;  that 
near  the  sun  a  part  of  the  cometary  matter  becomes  invisible,  having 
been  evaporated,  perhaps,  by  the  solar  heat,  just  as  a  cloud  of  fog 
might  be. 

The  change  is  especially  conspicuous  in  Encke's  comet.  When  this  body 
first  comes  into  sight,  at  a  distance  of  about  130  000000  miles  from  the 
sun,  it  has  a  diameter  of  nearly  300000  miles.  When  it  is  near  the  peri- 
helion, at  a  distance  from  the  sun  of  only  33  000000  miles,  its  diameter 
shrinks  to  12000  or  14000  miles,  the  volume  then  being  less  than  TTyt3^u  °^ 
what  it  was  when  first  seen.  As  it  recedes  it  expands,  and  resumes  its 
original  dimensions.  Other  comets  show  a  similar,  but  usually  less  strik- 
ing, change. 

716.  Dimensions  of  the  Nucleus. — This  has  a  diameter  ranging  in 
different  comets  from  6000  or  8000  miles  in  diameter  (Comet  III, 
1845)  to  a  mere  point  not  exceeding  100  miles.     Like  the  head,  it 
also  undergoes  considerable  and  rapid  changes  in  diameter,  though  its 
changes  do  not  appear  to  depend  in  any  regular  way  upon  the  comet's 


440  COMETS. 

distance  from  the  sun,  but  rather  upon  its  activity  at  the  time.    They 
are  usually  associated  with  the  development  of  jets  and  envelopes. 

717.  Dimensions  of  a  Comet's  Tail.  —  The  tail  of  a  large  comet,  as 
regards  simple  magnitude,  is  by  far  its  most  imposing  feature.     The 
length  is  seldom  less  than  10,000000  to  15,000000  miles  ;  it  frequently 
reaches  from  30,000000  to  50,000000,  and  in  several  cases  has  been 
known  to  exceed  100,000000.     It  is  usually  more  or  less  fan-shaped,  so 
that  at  the  outer  extremity  it  is  millions  of  miles  across,  being  shaped 
roughly  like  a  cone  projecting  behind  the  comet  from  the  sun,  and  more 
or  less  bent  like  a  horn.     The  volume  of  such  a  train  as  that  of  the 
comet  of  1882, 100,000000  miles  in  length,  and  some  200,000  miles  in 
diameter  at  the  comet's  head,  with  a  diameter  of  10,000000  at  its  ex- 
tremity, exceeds  the  bulk  of  the  sun  itself  by  more  than  8000  times. 

718.  The   Mass   of   Comets. — While   the   volume   of   comets   is 
enormous,  their  masses  appear  to  be  insignificant.     Our  knowledge 
in  this  respect  is,  however,  thus  far  entirely  negative;  that  is,  while  in 
many  cases  we  are  able  to  say  positively  that  the  mass  of  a  particular 
comet  cannot   have  exceeded  a  limit  which  can  be  named,  we  have 
never  been   able  to  fix  a  lower  limit  which  we  know  it  must  have 
reached ;    it  has  in  no  case  been  possible  to  detect  any  action  what- 
ever produced  by  a  comet  on  the  earth  or  any  other  body  of  the  plane- 
tary system,  from  which  we  can  deduce  the  comet's  mass ;  and  this, 
although   they  have  frequently  come  so  near  the   earth   and   other 
planets  that  their  own  orbits  have  been  entirely  transformed,  and  if 
their  masses  had  been  as  much  as  JOO^OTT  °^  *ne  earth's,  they  would 
have  produced  very  appreciable  effects  upon  the  motion  of  the  planet 
which  disturbed  them. 

Lexell's  comet  of  1770,  Biela's  comet  on  more  than  one  occasion,  and  sev- 
eral others,  have  come  so  near  the  earth  that  the  length  of  their  periods  of 
revolution  have  been  changed  by  the  earth's  attraction  to  the  extent  of 
several  weeks,  but  in  no  instance  has  the  length  of  the  year  been  altered 
by  a  single  second.  One  might  be  tempted  to  think  that  comets  were  pos- 
sessed of  matter  without  attracting  power ;  but  attraction  is  always  mutual, 
and  since  the  comets  move  according  to  the  law  of  gravitation,  and  them- 
selves suffer  perturbation  from  attraction,  there  is  no  escape  from  the  con- 
clusion that,  enormous  as  they  are  in  volume,  they  contain  very  little  matter. 
Some  have  gone  so  far  as  to  say  that  a  comet  properly  packed  could  be  car- 
ried about  in  a  hat-box  or  a  man's  pocket,  which,  of  course,  is  an  extravagant 
assertion.  The  probability  is  that  the  total  amount  of  matter  in  a  comet  of 
any  size,  though  very  small  as  compared  with  its  bulk,  is  yet  to  be  estimated 


DENSITY.  441 

at  many  millions  of  tons.  The  earth's  mass  (Art.  132, 4)  is  expressed  in  tons 
by  6  with  twenty-one  ciphers  following  (6000  millions,  of  millions,  of  millions 
of  tons).  A  body,  therefore,  weighing  only  one-millionth  as  much  as  the 
earth  would  contain  6000  millions  of  millions  of  tons,  which  is  very  nearly 
equal  to  the  mass  of  the  earth's  atmosphere. 

719.  The  late  Professor  Peirce  based  his  estimate  of  a  comet's 
mass  upon  the  extent  of  the  nebulous  envelope  which  it  carries  with 
it,  assuming  that  this  envelope  is  gaseous,  and  is  held  in  permanent 
equilibrium  by  the  attraction  of  solid  matter  in  and  near  the  nucleus  ; 
and  on  this  very  doubtful  assumption  he  came  to  the  conclusion  that 
the  matter  in  and  near  the  nucleus  of  an  average  comet  must  be 
equivalent  in  mass  to  an  iron  ball  as  much  as  100  miles  in  diameter. 
This  would  be  about  ^ooWo^  °^  ^ne  earth's  mass.     While  this  esti- 
mate is  not  intrinsically  improbable,  it  cannot,  however,  be  relied 
upon.     We  simply  do  not  know  anything  about  a  comet's  mass,  ex- 
cept that  it  is  exceedingly  small  as  compared  with  that  of  the  earth. 

720.  Density.  —  This  must  necessarily  be  almost  inconceivably 
small.    If  a  cornet  40000  miles  in  diameter  has  a  mass  equal  to  ^5  oV  OIF 
of  the  earth's  mass,  its  mean  density  is  a  little  less  than  ^o1^  of  that 
of  the  air  at  the  earth's  surface,  —  much  lower  than  that  of  the  best 
airpump  vacuum.     Near  the  centre  of  the  comet  the  density  would 
probably  be  greater  than  the  mean  ;  but  near  its  exterior  very  much 
less.     As  for  the  density  of  its  tail,  when  such  a  comet  has  one,  that, 
of  course,  must  be  far  lower  yet,  and  much  below  the  density  of  the 
residual  gas  left  in  the  best  vacuum  we  can  make  by  any  means 
known  to  science. 

This  estimate  of  the  density  of  a  comet  is  borne  out  by  the  fact 
that  small  stars  can  be  seen  through  the  head  of  a  comet  100000 
miles  in  diameter,  and  even  very  near  its  nucleus,  with  hardly  any 
perceptible  diminution  of  their  lustre.  In  such  cases  the  writer  has 
noticed  that  the  image  of  a  star  is  rendered  a  little  indistinct ;  and 
recent  observations  of  several  astronomers  have  shown  a  very  small 
apparent  displacement  of  the  star,  such  as  might  be  ascribed  to  a 
slight  refraction  produced  by  the  gaseous  matter  of  the  comet. 

Students  often  find  difficulty  in  conceiving  how  bodies  of  so  infinitesimal 
density  as  comets  can  move  in  orbits  like  solid  masses,  and  with  such 
enormous  velocities.  They  forget  that  in  a  vacuum  a  feather  falls  as  freely 
and  as  swiftly  as  a  stone.  Interplanetary  space  is  a  vacuum  far  more  per- 
fect than  any  airpump  could  produce,  and  in  it  the  rarest  and  most  tenuous 
bodies  move  as  freely  and  swiftly  as  the  densest. 


442  COMETS. 

721.  The  reader,  however,  must  bear  in  mind  that,  although  the 
mean  density  of  a  comet  (that  is,  the  quantity  of  matter  in  a  cubic 
mile)  is  small,  the  density  of  the  constituent  particles  of  a  comet  need 
not  necessarily  be  so.     The  comet  may  be  composed  of  small,  heavy 
bodies,  widely  separated,  and  there  is  some  reason  for  thinking  that  this 
is  the  case  ;   that,  in  fact,  the  head  of  a  comet  is  a  swarm  of  meteoric 
stones  ;  though  whether  these  stones  are  many  feet  in  diameter,  or  only 
a  few  inches,  or  only  a  few  thousandths  of  an  inch,  like  particles  of 
dust,  no  one  can  say.     In  fact,  it  now  seems  quite  likely  that  the 
greatest  portion  of  a  comet's  mass  is  made  up  of  such  particles  of 
solid  matter,  carrying  with  them  a  certain  quantity  of  enveloping  gas. 

722.  Light  of  Comets.  — There  has  been  much  discussion  whether 
these  bodies  shine  by  light  reflected  or  intrinsic.     The  fact  that  they 
become  less  brilliant  as  they  recede  from  the  sun,  and  finally  dis- 
appear while  they  are  in  full  sight  simply  on  account  of  faintness,1 
and  not  by  becoming  too  small  to  be  seen,  shows  that  their  light  is  in 
some  way  derived    from    the    sun.     The  further  fact  that  the  light 
shows  traces  of  polarization  also  indicates  the  presence  of  reflected 
sunlight.     But  while  the  light  of  a  comet  is  thus  in  some  way  attribu- 
table   to  the  sun's  action,  the  spectroscope  shows  that  it  does   not 
consist,  to  any  considerable  extent,  of  mere  reflected  sunlight,  like 
that  of  the  moon  or  a  planet. 

723.  If  a  comet  shone  by  mere  reflected  light,  or  by  any  light  the 
intensity  of  which  is  proportional  inversely  to  the   square   of  the  sun's 
distance  (as  would  naturally  be  the  case  if  the  light  were  excited  directly 
by  the  sun's  radiation,  and  proportional  to  it),  we  should  have  its  apparent 
brightness  at  any  time  equal  to  the  quantity         g,  in  which  D  and  A  are 

the  comet's  distances  from  the  sun  and  from  the  earth  respectively.  The 
brightness  of  a  comet  does,  in  fact,  generally  follow  this  law  roughly,  but 
with  many  and  striking  exceptions.  The  light  of  a  comet  often  varies 
greatly  and  almost  capriciously,  shining  out  for  a  few  hours  with  a  splendor 
seven  or  eight  fold  multiplied,  and  then  falling  back  to  the  normal  state  or 
even  below  it.  The  Pons-Brooks  comet  in  1883  and  Holmes'  comet  in  1892 
furnished  remarkable  instances  of  this  sort. 

724.  The  Spectra  of  Comets. — The   spectrum  of  most  comets 
consists  of  a  more  or  less  faint  continuous  spectrum  (which  may  be 

1  If  a  comet  shone  with  its  own  independent  light,  like  a  star  or  a  nebula, 
then,  so  long  as  it  continued  to  show  a  disc  of  sensible  diameter,  the  intrinsic 
brightness  of  this  disc  would  remain  unchanged :  it  would  only  grow  smaller  as  it 
receded  from  the  earth,  not  fainter. 


METALLIC   LINES   IN    SPECTRUM.  443 

due  to  reflected  sunlight,  though  it  is  usually  too  faint  to  show  the 
Fraunhofer  lines)  overlaid  by  three  bright  bands, — one  in  the  yellow, 
one  in  the  green,  and  the  third  in  the  blue.  These  bands  are  sharply 
defined  on  the  lower,  or  less  refrangible,  edge,  and  fade  out  towards 
the  blue  end  of  the  spectrum.  A  fourth  band  is  sometimes  visible 
in  the  violet.  The  green  band,  which  is  much  the  brightest  of  the 
three,  in  some  cases  is  crossed  by  a  number  of  fine,  bright  lines,  and 
there  are  traces  of  similar  lines  in  the  yellow  and  blue  bands.  JFhis 
spectrum  is  absolutely  identical  with  that  given  by  the  blue  base  of  an 
ordinary  gas  or  candle  flame y1  or  better,  by  the  blue  flame  of  a  Bun- 
sen  burner  consuming  ordinary  illuminating  gas.  Almost  beyond 
question  it  indicates  the  presence  of  some  gaseous  carbon  compound, 
a  hydrocarbon  or  cyanogen,  which  in  some  way  is  made  to  shine; 
either  by  a  general  heating  to  the  point  of  luminosity  (which  is  hardly 
probable),  or  by  electric  discharges  within  it,  or  by  local  heatings  due 
to  collisions  between  the  solid  masses  disseminated  through  the  gas- 
eous envelope ;  or  possibly  to  phosphorescence  due  to  the  action  of 
sunlight :  or  none  of  these  surmises  may  be  correct,  and  we  may 
have  to  seek  some  other  explanation  not  yet  suggested. 

It  is  not  at  all  certain  that  the  temperature  of  the  comet,  considered 
as  a  whole,  is  very  high.  Nor  will  it  do  to  suppose  that  because  the 
spectrum  reveals  the  presence  in  the  comet  of  gaseous  hydrocarbon, 
this  substance,  therefore,  composes  the  greater  part  of  the  comet's 
mass.  The  probability  is  that  the  gaseous  portion  of  the  comet  is 
only  a  small  percentage  of  the  whole. 

725,  Metallic  Lines  in  Spectrum.  —  When  a  comet  approaches 
very  near  to  the  sun,  as  did  Wells's  comet  in  1882,  and  a  few  weeks 
later  the  great  comet  of  that  year,  the  spectrum  shows  bright  metal- 
lic lines  in  addition  to  the  hydrocarbon  bands.     The  lines  of  sodium 
and  magnesium  are  most  easily  and  certainly  recognizable.     As  for 
the  other  lines  —  a  multitude  of  which  were  seen  by  Ricco  (of 
Palermo)  for  a  few  hours,  in  the  spectrum  of  the  great  comet  of 
1882  —  they  are  probably  due  to  iron;  though  that  is  not  certain, 
for  they  were  not  seen  long  enough  to  be  studied  thoroughly. 

726.  Anomalous  Spectra.  —  While  most  comets  show  the  hydro- 
carbon spectrum,  occasionally  a  different  spectrum  of  bands  appears. 
Fig.  198  shows  the  spectra  of  three  comets  compared  with  the  solar 
spectrum  and  with  that  of  hydrocarbon  gas. 

1  It  is  not  the  spectrum  of  carbon  monoxide,  CO,  as  has  been  stated  by  Flam- 
marion  and  others,  though  there  is  some  evidence  of  the  presence  of  that  sub- 
stance as  a  subordinate  constituent. 


444  COMETS. 

The  first,  the  spectrum  of  Tebbutt's  comet  of  1881,  is  the  usual  one. 
The  other  two  are  unique.  Brorsen's  comet,  at  its  later  returns,  showed  the 
ordinary  comet  spectrum,  and  it  might  perhaps  be  considered  possible  that 
an  error  was  made  in  fixing  the  position  of  the  bands  at  the  first  observa- 
tion. But  the  peculiar  spectrum  of  comet  C,  1877,  hardly  permits  such 
an  explanation.  It  was  observed  at  Dunecht  on  the  same  night,  by  the 
same  observers  and  with  the  same  spectroscope,  as  another  comet  which 


FIG.  198.  — Comet  Spectra. 

(For  convenience  in  engraving,  the  dark  lines  of  the  solar  spectrum  in  the  lowest  strip  of  the 
figure  are  represented  as  bright.) 

gave  the  usual  spectrum ;  so  that  in  this  case  it  hardly  seems  possible  that 
the  anomalous  result  can  be  a  mistake,  though  the  spectrum  itself  as  yet 
remains  unidentified  and  unexplained. 

Holmes's  comet  of  1892,  unlike  any  other  yet  observed,  gave  a  simply  con- 
tinuous spectrum,  without  perceptible  markings  either  bright  or  dark. 

It  is  maintained  by  Mr.  Lockyer  that  the  spectrum  of  a  comet  changes  as 
it  varies  its  distance  from  the  sun,  the  bands  altering  in  appearance  and 
shifting  their  position.  But  the  evidence  of  this  is  not  yet  conclusive. 

It  is  certainly  remarkable  that  comets,  coming  as  they  do  from  widely 
separated  regions  of  space,  show  so  little  variety  in  their  spectra :  a  priori  we 
should  expect  difference  rather  than  resemblance. 

727.  Development  of  Jets  and  Envelopes,  —  When  a  comet  is  first 
seen  at  a  great  distance  from  the  sun  it  is  ordinarily  a  mere  roundish, 
hazy  patch  of  faint  nebulosity,  a  little  brighter  near  the  centre. 


DEVELOPMENT   OF   JETS   AND   ENVELOPES.  445 

As  the  cornet  draws  near  the  sun  it  brightens,  and  the  central  con- 
densation becomes  more  conspicuous  and  sharply  defined,  or  star-like. 


FIG.  199.  —  Head  of  Donati's  Comet,  Oct.  5,  1858.     (Bond.) 

Then,  on  the  side  next  the  sun,  the  newly  formed  nucleus  begins  to 
emit  jets  and  streamers  of  light,  or  to  throw  off  more  or  less  sym- 
metrical envelopes,  which  fol- 
low each  other  concentrically 
at  intervals  of  some  hours,  ex- 
panding and  growing  fainter 
as  they  ascend,  until  they  are 
lost  in  the  general  nebulosity 
which  forms  the  head.  Dur- 
ing these  processes  the  nucleus 
continually  changes  in  bril- 
liancy and  magnitude,  usually 
growing  smaller  and  brighter 
just  before  the  liberation  of 
each  envelope.  When  jets  are 
thrown  off,  the  nucleus  seems 

tO      OSCillate,     moving      Slightly       ^o.  200. -Tebbutt's  Comet,  1881.     (Common.) 

from  side  to  side ;  but  no  evidences  of  a  continuous  rotation  have 
ever  been  discovered.     The  two  figures,  199  and  200,  represent  the 


446 


COMETS. 


heads  of  two  comets  which  behaved  quite  differently.  Fig.  199  is 
the  head  of  Donati's  comet  as  seen  on  Oct.  5,  1858.  This  comet 
was  characterized  by  the  quiet,  orderly  vigor  of  its  action.  It  did 
very  little  that  was  anomalous  or  erratic,  but  behaved  in  all  respects 
with  perfect  propriety.  The  system  of  envelopes  in  the  head  of  this 
comet  was  probably  the  most  symmetrical  and  beautiful  ever  seen. 
Fig.  200  is  from  a  drawing  by  Common  of  the  head  of  Tebbutt's 
comet  in  1881.  This  comet,  on  the  other  hand,  was  always  doing 
something  outre,  throwing  off  jets,  breaking  into  fragments,  and,  in 
fact,  continually  exhibiting  unexpected  phenomena. 

728.  Formation  of  the  Tail.  —  The  material  which  is  projected 
from  the  nucleus  of  the  comet,  as  if  repelled  by  it,  is  also  repelled 
by  the  sun,  and  driven  backward,  still  luminous,  to  form  the  train.  (At 

least,  this  is  the  appearance.)  Fig. 
201  shows  the  manner  in  which  the 
tail  is  thus  supposed  to  be  formed.1 
The  researches  of  Bessel,  Norton, 
and  especially  the  late  investiga- 
tions of  the  Russian  Bredichin,  have 
shown  that  this  theory  —  that  the 
tail  is  composed  of  matter  repelled 
by  both  the  comet  and  the  sun  — 
not  only  accounts  for  the  phenomena 
in  a  general  way,  but  for  almost  all 
the  details,  and  agrees  mathemat- 
ically with  the  observed  position 
and  magnitude  of  the  tail  on  differ- 
ent dates. 


To  Sun 


FIG.  201. 


Formation  of  a  Comet's  Tail  by  Matter 
expelled  from  the  Head. 


The  repelled  particles  are  still  subject  to  the  sun's  gravitational  attraction, 
and  the  effective  force  acting  upon  them  is  therefore  the  difference  between 
the  gravitational  attraction  and  the  electrical  (?)  repulsion.  This  difference 
may  or  may  not  be  in  favor  of  the  attraction,  but  in  any  case,  the  sun's 
attracting  force  is,  at  least,  lessened.  The  consequence  is  that  those  repelled 


1  Other  theories  of  comets'  tails  have  been  presented,  and  have  had  a  certain 
currency,  —  theories  according  to  which  the  tail  is  a  mere  "luminous  shadow"  of 
the  comet,  so  to  speak,  or  a  swarm  of  meteors.  But  all  these  theories  break 
down  in  the  details.  They  fail  to  account  for  the  phenomena  of  jets,  envelopes, 
etc.,  in  the  head  of  the  comet,  and  they  furnish  no  mathematical  determinatioK 
of  the  outlines  and  curvature  of  the  tail. 


CURVATURE    OF    THE    TAIL. 


447 


particles,  as  soon  as  they  get  a  little  away  from  the  comet,  begin  to  move 

around  the  sun  in  hyperbolic 1  orbits  which 

lie  in  the  plane  of  the  comet's  orbit,  or 

nearly  so,  and  are  perfectly  amenable  to 

calculation. 

The  tail  is  simply  an  assemblage  of  these 
repelled  particles,  and,  according  to  theory, 
ought,  therefore,  to  be  a  sort  of  flat,  hollow, 
horn-shaped  cone,  as  represented  by  Fig. 
202,  open  at  the  large  end,  and  rounded 
and  closed  at  the  smaller  one,  which  con- 
tains the  nucleus. 


FIG.  202. 
A  Comet's  Tail  as  a  Hollow  Cone. 


729.  Curvature  of  the  Tail.  —  The  cone  is  curved  as  shown, 
because  the  particles  repelled  still  retain  their  original  orbital  motion, 
so  that  they  will  not  be  arranged  along  a  straight  line  drawn  from 


FIG.  203.  —A  Comet's  Tail  at  Different  Points  in  its  Orbit  near  Perihelion. 

the  sun  through  the  comet,  but  along  a  curve  convex  to  the  direction 
of  the  comet's  motion  ;  but  the  stronger  the  repulsion,  the  less  will 
be  the  curvature.  Fig.  203  shows  how  the  tail  ought  to  lie  as  the 

1  Referring  to  the  formula  for  the  semi-major  axis  of  an  orbit,  viz., 

U*     \  (Art.  429), 


we  see  that  a  repulsive  force  acting  from  the  sun  diminishes  U  (which  measures 
the  sun's  attraction),  and  the  consequence  is  that  if  the  unrepelled  particles  are 
describing  a  parabola  (in  which  case  U2=  T2),  then  for  the  repelled  particles  the 
denominator  will  become  negative  (  U  having  been  made  smaller  than  V  by  the 
repulsive  action),  and  thus  a  will  also  become  negative,  so  that  the  orbit  for  a 
repelled  particle  will  be  a  hyperbola. 


448  COMETS. 

comet  rounds  the  perihelion  of  its  orbit.  According  to  this  theory, 
the  tail  should  be  hollow,  and  in  the  case  of  comets  when  at  their 
brightest  it  usually  seems  to  be  so,  the  centre  being  darker  than  the 
edges, 

730.  The  Central  Stripe  in  a  Comet's  Tail.  —  Very  often,  there 
is  a  peculiar  straight,  dark  stripe  through  the  axis  of   the    tail   as 
shown  in  Figs.  199  and  200  of  the  head  of  Donati's  and  Tebbutt's 
comets.     It  might  be  mistaken  for  the  shadow  of  the  nucleus  if  it 
were  pointed  exactly  away  from  the  sun  ;  but  it  is  not,  usually  making 
an  angle  of   several  degrees  with  the  direction   of  a   true  shadow. 
Sometimes,  however,  and  not  very  unfrequently,  the  tail  has  a  bright 

centre  instead  of  a  dark  one,  perhaps  on 
account  of  the  feebleness  of  the  comet's 
own  repulsive  action  ;  in  fact,  this  seems 
to  be  usually  the  case  when  the  comet 
has  reached  a  considerable  distance  from 
the  sun  in  receding  from  it,  and  often  it 
is  so  when  the  comet  is  approaching  the 

FIG.  204.  r  . 

sun,  but  is  still  remote,  as  in  the   case 

Bright  Centred  Tail  of  Coggia's 

Comet.    June,  1874.  of  Coggia  s  comet  shown  in  Fig.  204. 

In  such  cases  the  tail  is  generally  faint 

and  ijl-defined  at  the  edge,  with  a  central  spine  of  light,  and  in  some 
cases  it  becomes  apparently  a  mere  slender  ray,  of  less  diameter 
than  the  head  of  the  comet  itself.  This,  however,  is  unusual. 
The  explanation  of  this  kind  of  tail  requires  a  slight  modification  of 
the  theory,  so  far  as  to  admit  that  the  particles  at  first  repelled  by 
the  front  of  the  comet  are  afterwards  attracted  by  it,  though  still 
repelled  by  the  sun. 

731.  Tails  of  Three  Different  Types.— Bredichin  has  found  that 
the  tails  of  comets  may  be  grouped  under  three  types  :  — 

1.  The  long,  straight  rays.  They  are  formed  of  matter  upon  which 
the  sun's  repulsive  action  is  from  twelve  to  fifteen  times  as  great  as 
the  gravitational  attraction,  so  that  the  particles  leave  the  comet  with 
a  relative  velocity  of  at  least  four  or  five  miles  a  second ;  and  this 
velocity  is  continually  increased  as  they  recede,  until  at  last  it  becomes 
enormous,  the  particles  travelling  several  millions  of  miles  in  a  day. 
The  straight  rays  which  are  seen  in  the  figure  of  the  tail  of  Donati's 
comet,  tangential  to  the  tail,  are  streamers  of  this  first  type ;  as  also 
was  the  enormous  tail  of  the  comet  of  1861. 


TAILS    OF   THREE   DIFFERENT   TYPES. 


449 


2.  The   second   type   is   the  curved,   plume-like   train,  like  the 
principal  tail  of  Donati's  comet.     In  this  type  the  repulsive  force 
varies  from  2.2  times  gravity  (for  the  particles  on  the  convex  edge  of 
the  tail)  to  half  that  amount  for  those  which  form  the  inner  edge. 
This  is  by  far  the  most 

common  type  of  come- 
tary  train. 

3.  A  few  comets  show 
tails  of  the  third  type, 
—  short,  stubby  brush- 
es violently  curved,  and 
due  to  matter  of  which 
the    repulsive    force    is 
only  a  fraction  of  grav- 
ity, —  from  y1^  to  l. 

732.  According  to 
Bredichin,  the  tails  of  the 
first  type  are  probably 
composed  of  hydrogen, 
those  of  the  second  type 
of  some  hydrocarbon  ga$, 
and  those  of  the  third  of 
iron  vapor,  with  probably 
an  admixture  of  sodium 
and  other  materials. 

There  has  been  no  op- 
portunity since  Bredichin 
published  this  result  to 
test  the  matter  spectro- 
scopically  for  tails  of  the 
first  and  third  types,  by 
looking  for  the  lines  of 
hydrogen  and  iron.  The 
hydrogen  tails  are  almost 
always  very  faint,  and  the 
tails  of  the  third  class  are  uncommon.  Tails  of  the  second  type,  which  are 
brightest  and  most  usual,  do  show  a  hydrocarbon  spectrum  throughout  their 
entire  length,  and  so  far  confirm  his  view. 

The  reason  for  this  conclusion  of  Bredichin  is  that  he  supposes  the 
repulsive  force  to  be  a  surface  action,  the  same  for  equal  surfaces  of  any  kind 
of  matter ;  the  effective  accelerating  force,  therefore,  measured  by  the  velocity 
it  would  produce,  would  depend  upon  the  ratio  of  surface  to  mass  in  the 
particles  acted  upon,  and  so,  in  his  view,  should  be  inversely  proportional 


FIG.  205.  —  Bredichin's  Three  Typea  of  Cometary  Tails. 


450  COMETS. 

to  their  molecular  weights.  Now  the  molecular  weights  of  hydrogen,  of 
hydrocarbon  gases,  and  of  the  vapor  of  iron,  bear  to  each  other  just  about 
the  required  proportion. 

733.  Nature  of  the  Repulsive  Force.  —  As  to  this,  our  knowledge 
is  still  imperfect,  but  the  remarkable  discoveries  made  since  1890 
indicate  that  repulsions  must  be  active  wherever  conditions  exist 
like  those  at  the  surface  of  the  sun.     Phenomena  must  necessarily 
there  appear  resembling  those  exhibited  in  our  laboratories  when 
bodies  at  high  temperature,  or  under  powerful  electrical  excitement, 
or  composed  of  such  peculiar  substances  as  radium  and  its  congeners, 
project  into  the  rarefied  medium  around  them  "ions"  and  "corpus- 
cles "  of  various  kinds  and  velocities.     Indeed,  as  Maxwell  pointed 
out  thirty  years  ago,  there  must  also  be,  according  to  his  electro- 
magnetic theory  of  light  (now  almost  universally  accepted),  a  pres- 
sure exerted  by  light-waves  upon  all  electro-conducting  masses  upon 
which  they  impinge :  a  force  very  minute  as  compared  with  gravity 
upon  masses  larger  than  pin-heads,  but  greatly  exceeding  it  for  par- 
ticles of  the  size  of  light-waves.     This  light-pressure,  long  vainly 
sought  by  experiment,  has  at  last  (1901-2)  been  detected  and  meas- 
ured by  Lebedew  in  Moscow,  and  by  Nichols  and  Hull  in  our  own 
country.     We  are  no  longer  at  a  loss  to  account  for  the  sun's  repul- 
sion upon  the  materials  of  the  corona  and  of  comets'  tails,  but  only 
to  determine  in  just  what  manner  the  different  forces  cooperate. 

A  singular  theory  has  been  proposed  by  Zenker,  that  the  repulsion  is 
due  to  the  reaction  produced  by  rapid  evaporation  on  the  surface  of  the 
little  solid  and  liquid  particles  of  which  he  supposed  a  comet  to  consist :  this 
evaporation  would,  of  course,  be  most  rapid  on  the  side  of  the  particles  next 
the  sun,  and  would  cause  a  recoil  in  a  manner  analogous  to  that  by  which 
the  so-called  spheroidal  state  of  liquids  is  produced  on  a  heated  surface. 
Ranyard  has  suggested  that  the  cometary  particles  may  consist  principally 
of  minute  liquid  drops  or  frozen  "hail-stones"  of  certain  hydrocarbons 
which  evaporate  rapidly  at  a  very  low  temperature  (such  as  rhigoline, 
naphtha,  and  their  congeners). 

734.  State  of  the  Matter  composing  the  Tail.  —  This  also  is  a  sub- 
ject of  speculation  rather  than  of  knowledge.     Perhaps  the  simplest 
supposition  is  that  we  have  to  do  with  gaseous  matter  rarefied  even 
beyond  the  limits  of  the  gas  contained  in  Crookes's  tubes,  —  so  rare- 
fied that  since  its  molecules  no  longer  suffer  frequent  collisions  with 
each  other,  it  has  thus  lost  all  the  peculiar  mechanical  characteris- 
tics of  a  gaseous  mass,  and  become  a  mere  cloud  of  separate  parti- 


WHAT   BECOMES   OF  MATTER   THROWN  OFF.  451 

cles,  each  particle  consisting,  however,  of  but  a  single  molecule, 
Spectroscopically  such  a  cloud  would  still  be  gaseous,  but  from  a 
mechanical  point  of  view  extremes  would  have  met,  and  this  most 
tenuous  gas  would  have  become  a  cloud  of  finely  powdered  solid. 

735.  What  becomes  of  the  Matter  thrown  off  in  Comets'  Tails.— 
To  this  we  have   no    certain  answer  at  present ;   but  if  the    theory 
which  has  been  stated  is  true,  it  is  clear  that  most  of  the  matter  so 
repelled  from  comets  can  never  be  re-gathered  by  the  nucleus,  but 
must  be  dissipated  in  space. 

Whenever  a  planet  meets  any  of  the  particles,  it  picks  them  up,  of  course, 
as  it  picks  up  meteors ;  and  Newton  long  ago  suggested,  what  has  of  late 
been  forcibly  dwelt  upon  by  Dr.  Sterry  Hunt,  that  in  this  way  the  atmos- 
pheres of  the  planets  may  be  supplied  with  material  to  take  the  place  of  the 
carbon  which  has  been  absorbed  and  fixed  by  the  processes  of  crystallization 
and  of  life.  Otherwise  it  would  seem  that  the  processes  now  going  on  upon 
the  earth's  surface  must  necessarily  in  the  course  of  time  deprive  the  atmos- 
phere of  all  its  carbonic  acid. 

If  this  view  is  correct,  it  follows  that  such  comets  as  have  tails  lose  a 
portion  of  their  substance  every  time  that  they  visit  the  sun.  It  is  quite  con- 
ceivable, also,  that  the  processes  by  which  light  is  excited  in  the  head  of  a 
comet  may  use  up  and  render  unfit  for  future  shining,  a  portion  of  its 
material ;  so  that,  as  a  periodic  comet  grows  old,  it  may  become  both  smaller 
and  less  luminous,  until  finally  it  ceases  to  be  observable. 

736.  Anomalous  Tails  and  Streamers.  — It  is  not  very  unusual  for 
comets  to  show  tails  of  two  different  types  at  the  same  time,  as,  for 
instance,  Donati's  comet.     But  occasionally  stranger  things  happen, 
and  the  great  comet  of  1744  is  reported  to  have  had  six  tails  diverg- 
ing like  a  fan.     Winnecke's  comet  of  1877  threw  out  a  tail  laterally, 
making  an  angle  of  about  60°  with  the  normal  tail,  and  having  the 
same  length, — about  1°.     Pechule's  comet  of  1880  (a  small  one), 
besides  the  normal  tail,  had  another  of  about  the  same  dimensions 
directed  straight  towards  the  sun :  streamers  of  considerable  length 
so  directed  are  not  very  infrequent.     The  great  comet  of  1882  pre- 
sented a  number  of  peculiarities,  which  will  be  mentioned   in  the 
more  particular  description  of  that  body,  which  is  to  follow.     Most 
of  these  anomalies  are  as  yet  entirely  unexplained. 

737.  Nature  of  Comets.  — It  is  obvious  from  what  has  been  said 
that  we  have  little  certain  knowledge  on  this  subject ;  but  perhaps  on 
the  whole  the  most  probable  hypothesis  is  the  one  which  has  been 


452  COMETS. 

hinted  at  repeatedly,  —  that  a  comet  is,  as  Professor  Newton  ex- 
presses it,  nothing  but  a  "  sand-bank  " ;  i.e.*  a  swarm  of  solid  parti- 
cles of  unknown  size  and  widely  separated  (say  pin-heads  several 
hundred  feet  apart),  each  particle  carrying  with  it  an  envelope  of 
gas,  largely  hydrocarbon,  in  which  gas  light  is  produced,  either  by 
electric  discharges  between  the  particles,  or  by  some  other  light- 
evolving  action1  due  to  the  sun's  influence. 

This  hypothesis  derives  its  chief  plausibility  from  the  modern  dis- 
covery of  the  close  relationship  between  meteors  and  comets,  to  be 
discussed  in  the  next  chapter. 

738.  Origin  of  Periodic  Comets. — It  is  obvious  that  the  comets 
which  move  in  parabolic  orbits  are,  as  has  been  said  already,  mere 
visitors  to  the  solar  system,  and  not  citizens  of  it :  but  as  to  those 
which   now  move   in  elliptical  orbits   around   the   sun,  returning  as 
regularly  as  planets,  it  is  a  question  whether  we  are  to  regard  them 
as  native-born,  or  only  as  naturalized.     Did  they  originate   in  the 
system,  or  are  they  captives? 

739.  Planets'  Families  of  Comets.  —  It  is  quite  clear  that  in  some 
way  or  other  many  of  them  owe  their  present  status  in  the  system  to 
Jupiter,  Saturn,  and  the  other  planets.    In  Article  703  we  called  atten- 
tion to  the  fact  that,  without  exception,  all  the  short-period  comets 
(i.e.,  those  having  periods  ranging  from  three  to  eight  years),  pass 
very  near  to  Jupiter's  orbit  at  some  point  in  their  paths ;  and  they 
are  now  recognized  and  spoken  of  as  Jupiter's  "family"  of  comets,  — 
about  thirty  of  them  are  reckoned  at  present,  their  number  having 
been  considerably  increased  by  the  discoveries  of  the  last  few  years. 

Fourteen  of  them  have  already  been  observed  at  two  or  more  returns,  and 
two  or  three  more  will  probably  be  reobserved  very  soon.  The  others  have 
failed  to  be  seen  a  second  time  either  by  pure  accident  or  on  account  of  un- 
favorable position,  or  they  may  have  suffered  the  same  mysterious  fate  as 
Biela's  comet  (Art.  745),  and  disappeared. 

1  Some  have  ascribed  the  light  to  the  collisions  between  the  little  stones  of 
which  they  assume  the  comet  to  be  made  up,  forgetting  that,  although  the  abso- 
lute velocity  of  the  comet  is  extremely  great,  the  relative  velocities  of  its  con- 
stituent masses  with  reference  to  each  other  must  be  very  slight  —  far  too  small 
apparently  to  account  for  any  considerable  rise  of  temperature  or  evolution  of 
light  in  that  way.  It  is  perhaps  worth  considering  whether  gases  in  the  mass  may 
not  become  sensibly  luminous  at  a  much  lower  temperature  than  has  usually 
been  supposed.  It  would  seem  not  improbable  a  priori  that  at  every  temperature, 
radiations  of  every  wave-length  must  be  emitted  in  some  degree ;  i.e.,  that  at  any 
temperature  above  the  absolute  zero  no  body  is  absolutely  non-luminous. 


ORIGIN   OF   COMETS.  453 

Similarly,  Saturn  is  at  present  credited  with  two  comets,  one  of 
which  is  Tuttle's  comet,  given  in  the  catalogue  of  periodic  comets. 
Uranus  stands  sponsor  for  three,  —  one  of  them  Tempel's  comet, 
which  is  very  interesting  in  its  relation  to  the  November  meteors,  and 
was  expected  back  in  1900,  but  failed  to  appear.  Finally,  Neptune 
has  a  family1  of  six.  Halley's  comet  is  one  of  them,  and  two  of  the 
others  have  been  observed  for  a  second  time  since  1880 ;  the  other 
three  are  not  due  for  some  years  to  come. 

740.  Origin  of  Comets:  the  " Capture"  Theory.  —  The  generally 
accepted  theory  as  to  the  origin  of  these  comet  families  is  that  the 
comets  which  compose  them  have  been  captured  by  the  planets  to 
which  they  now  belong.  This  was  first  suggested  by  Laplace. 

A  comet  entering  the  system  from  an  infinite  distance,  and  moving 
in  a  parabolic  orbit,  when  it  comes  near  a  planet  will  be  either 
accelerated  or  retarded.  If  accelerated,  its  orbit  becomes  hyperbolic, 
so  that  it  never  returns  for  a  second  observation.  If,  on  the  other 
hand,  it  is  retarded,  the  orbit  becomes  elliptical,  and  the  comet  will 
return  at  regular  intervals,  moving  in  a  path  which,  of  course,  always 
passes  through  the  point  where  the  disturbance  took  place. 

It  is  true,  as  Mr.  Proctor  has  pointed  out,  that  the  attraction  of 
Jupiter,  huge  as  is  his  mass,  could  not  at  one  effort  transform  a  para- 
bolic orbit  into  an  orbit  so  small  as  that,  say,  of  Biela's  comet.  But 
it  is  not  necessary  that  the  thing  should  be  done  at  one  effort.  The 
comet's  orbit  lies  near  to  Jupiter's,  and  after  a  lapse  of  time,  Jupiter 
and  the  comet  will  be  sure  to  come  alongside  again  :  the  comet  may 
then  be  sent  into  a  hyperbolic  or  parabolic  orbit,  —  the  chances  for 
such  a  result  are  nearly  even ;  —  but  it  may  also  have  its  velocity  a 
second  time  diminished,  and  its  orbit  made  still  smaller;  and  this  may 
be  done  over  and  over  again  unlimitedly,  until  the  aphelion  of  the 
comet  falls  at  such  a  distance  within  the  orbit  of  Jupiter  that  the 
planet  is  no  longer  able  to  disturb  it  seriously.  Given  time  enough, 
and  comets  enough,  for  Jupiter  to  work  upon,  and  the  final  result 
would  necessarily  be  a  comet-f amity  such  as  really  exists,  with  the 
aphelia  of  their  orbits  near  to  the  orbit  of  Jupiter,  and  periods 
roughly  half  his  own.  But  it  must  be  frankly  admitted  that  the 
extent  of  time,  and  the  quantity  of  cometary  material  demanded,  are 
enormous. 

1  Comet-Families  must  be  carefully  distinguished  from  Comet-Groups.  The 
comets  of  a  single  "group"  all  have  orbits  nearly  coincident,  at  least  in  the 
region  near  the  sun.  The  orbits  of  a  "family  "  have  no  special  resemblance  to 
each  other  except  in  period,  and  in  near  approach  to  the  orbit  of  the  planet  to 
which  they  belong. 


454  COMETS. 

740*.  This  "  capture  theory  "  has  recently  received  a  fine  illustration  in 
the  case  of  a  little  comet,  1889  V,  discovered  by  Brooks.  It  was  very  soon 
found  to  be  a  member  of  Jupiter's  "  family  "  with  a  period  of  about  7  years, 
and  on  careful  investigation  Dr.  S.  C.  Chandler,  of  Cambridge  (U.  S.),  ascer- 
tained that  in  1886  the  comet  and  the  planet  had  come  very  near  each  other, 
and  that  as  a  consequence  the  comet's  orbit  must  have  been  completely 
transformed,  the  previous  orbit  having  been  much  larger,  with  a  period  of 
about  27  years.  Now  the  researches  of  Laplace,  and  more  recently  of 
Leverrier,  had  shown  that  Lexell's  lost  comet  of  1770  (which,  according  to 
the  observations  then  made,  had  a  period  of  only  5£  years,  but  was  never  seen 
again)  (Art.  718)  was  removed  from  our  range  of  observation  by  a  similar 
encounter  with  Jupiter  in  1779,  which  transformed  the  then  small  orbit  into 
one  much  larger.  Dr.  Chandler  showed  that,  so  far  as  could  be  determined 
from  the  observations  then  available,  it  appeared  not  only  possible,  but 
extremely  probable,  that  Brooks's  comet  was  identical  with  Lexell's  ;  and 
for  some  time  it  was  generally  referred  to  as  the  Lexell-Brooks  comet.  Later 
researches,  however,  by  Dr.  C.  L.  Poor,  of  Baltimore,  based  on  more  extended 
observations,  while  confirming  the  closeness  of  approach  to  Jupiter  in  1886, 
throw  great  doubt  upon  the  absolute  identity  of  the  Brooks  comet  with 
Lexell's,  and  make  it  more  probable  that  the  two  are  related  merely  as  origi- 
nally members  of  the  same  "  comet-group  "  (Art.  705) ;  a  conclusion  strength- 
ened by  Swift's  discovery  of  comet  1895  II,  which,  according  to  Schulhof, 
meets  the  conditions  of  identity  with  Lexell's  even  better  than  Brooks's. 

Comet  1889  V  returned  again  in  1896,  having  been  found  in  June  very 
near  the  place  predicted  by  Poor.  It  was  faint  and  not  well  situated,  but 
on  the  whole  the  observations  decidedly  favor  Poor's  conclusion  that  it  is 
not  identical  with  Lexell's. 

In  1889  the  comet  was  observed  by  Barnard  at  the  Lick  Observatory  to 
be  double,  and  the  two  parts  were  slowly  separating  at  a  rate,  which,  reckoned 
backward,  would  indicate  that  the  disruption  had  occurred  in  1886  when  the 
planet  was  close  to  Jupiter.1  According  to  the  computations  of  Dr.  Poor,  the 
comet  then  actually  passed  between  the  surface  of  the  planet  and  the  orbit 
of  the  first  satellite. 

741.  The  "  Ejection  "  Theory.  —  Mr.  Proctor  has  suggested,  and 
vigorously  defended,  a  very  different  theory,  —  that  comets  are  masses  of  matter 
which  have  been  thrown  off  from  the  heavenly  bodies  by  eruptions  of  some  sort; 
that  the  comets  of  Jupiter's  family,  for  instance,  once  formed  a  portion  of 
its  mass,  and  were  at  some  times  ejected  with  a  velocity  sufficient  to  set  them 
free  in  space ;  and  that  many  of  the  parabolic  comets  may  have  been  sim- 
ilarly ejected  from  our  own,  or  from  other  suns.  The  main  difficulty  with  this 
theory  is  that  there  is  no  evidence  of  the  necessary  eruptive  energy  in  Jupiter, 
or  in  any  of  the  planets.  A  body  would  have  to  leave  the  upper  surface  of 
Jupiter's  atmosphere  with  a  velocity  exceeding  thirty-five  miles  a  second. 

1  See  note  at  end  of  chapter,  on  the  disintegration  of  comets. 


REMARKABLE   COMETS.  455 

It  cannot  be  said,  however,  that  there  is  any  special  mechanical  difficulty 
in  supposing  that  some  of  the  parabolic  comets  may  owe  their  origin  to 
eruptions  from  distant  suns.  Our  own  sun  unquestionably  sometimes  ejects 
clouds  of  matter  (in  the  form  of  the  solar  prominences)  with  enormous 
velocity,  perhaps  in  some  cases  sufficient  to  send  them  off  into  space.  But 
so  far  as  we  can  make  out  from  the  spectroscopic  evidence,  the  material  of 
comets  is  entirely  different  from  that  of  the  prominences. 

741*.  The  Home  of  the  Comets.  —  There  are  difficulties  connected 
with  the  theory  that  cornets  come  to  us  from  interstellar  space, 
chiefly  depending  upon  the  now  certain  fact  that  the  solar  system  is 
travelling  at  the  rate  of  several  miles  a  second  (Art.  806),  and  that 
therefore  comets  composed  of  matter  met  by  us  ought  to  have  a 
relative  velocity  so  great  as  to  produce  numerous  hyperbolic  orbits, 
whereas  we  find  few  such,  if  any.  Then  too  there  ought  to  be  a 
marked  concentration  of  the  axis  of  cometary  orbits  near  the  direc- 
tion of  the  solar  motion.  While  the  investigations  of  the  late 
Professor  Newton,  of  New  Haven,  partially  relieve  the  objections, 
other  astronomers  still  feel  them ;  and  it  has  been  suggested  that 
>"the  home  of  the  comets/7  as  Professor  Peirce  called  it,  maybe 
a  mass  or  shell  of  nebulous  matter  accompanying  the  system  in  its 
motion,  and  surrounding  it  at  a  distance  some  thousands  of  times 
greater  than  the  earth's  distance  from  the  sun,  but  still  much  closer 
than  the  nearest  stars  (for  of  the  stars  our  next  neighbor,  a 
Centauri,  is  275000  times  remoter  than  the  sun).  A  comet  starting 
initially  from  such  a  " shell7'  at  a  distance  of  10000  astronomical 
units  would  move  in  a  long  ellipse  with  a  period  of  a  million  years 
and  no  human  observation  could  detect  its  deviation  from  an  exact 
parabola.  Moreover,  if  comets  came  to  us  indiscriminately  from  all 
portions  of  this  nebulosity,  their  orbits  would  lie  indiscriminately 
in  all  directions,  and  in  every  plane,  just  as  we  find  them.  But  as 
yet  we  have  no  direct  evidence  of  any  such  comet-dropping  envelope. 

742.  Remarkable  Comets.  —  (1)  Halle^s  Comet.  This  was  the 
first  periodic  comet  whose  return  was  predicted.  Halley  based  his 
prediction  upon  the  fact  that  he  found  its  orbit  in  1682  to  be  nearly 
identical  with  those  of  the  comets  of  1607  and  1531,  which  had  been 
carefully  observed  by  Kepler  and  Apian ;  and  he  also  found  records 
of  the  appearance  of  great  comets  in  1456,  in  1301,  in  1145  and 
1066,  which  would  correspond  as  regards  the  time-intervals  concerned, 
though  data  were  wanting  for  an  accurate  calculation  of  their  orbits. 
He  noticed,  of  course,  that  the  two  intervals  between  1531  and  1607, 


456  COMETS. 

and  between  1607  and  1682  were  not  quite  equal ;  but  he  had  sagac- 
ity enough  to  see  that  the  differences  were  no  greater  than  might  be 
accounted  for  by  the  attractions  of  Jupiter  and  Saturn. 

The  theory  of  perturbation  was  not  then  sufficiently  developed  to  make  it 
possible  to  compute  with  precision  just  what  the  effect  would  be  upon  the 
next  return  of  the  comet,  but  he  saw  that  the  action  of  Jupiter  would 
retard  it,  and  he  accordingly  fixed  upon  the  early  part  of  1759  as  the  time 
at  which  the  comet  might  be  expected.  Before  that  date,  however,  math- 
ematics had  so  advanced  that  the  necessary  calculations  could  be  made. 
Clairaut,  as  the  result  of  a  most  laborious  investigation,  fixed  upon  April  13 
for  the  perihelion  passage ;  but  in  publishing  his  result,  he  remarked  that 
it  might  easily  be  a  month  out  of  the  way  owing  to  the  uncertainty  as  to 
the  masses  of  the  planets,  and  the  possible  action  of  undiscovered  planets 
beyond  Saturn  (Uranus  and  Neptune  were  then  unknown).  The  comet 
actually  came  to  perihelion  on  March  13.  At  this  return  it  was  best  seen 
in  the  southern  hemisphere,  and  at  one  time  had  a  tail  nearly  50°  long.  At 
its  next  return,  in  1835,  it  came  to  the  predicted  time  within  two  days.  It 
did  not  appear  on  this  occasion  as  an  extremely  brilliant  comet,  but  was 
reasonably  conspicuous,  with  a  tail  of  the  first  type  (hydrogen)  about  15° 
in  length. 

Its  next  return  will  occur  in  May,  1910,  but  the  necessary  calculations 
have  not  yet  been  made  to  determine  the  precise  date  with  accuracy. 

The  most  remarkable  of  its  earlier  appearances  were  in  1066  and  1456. 
The  comet  of  1066  figures  on  the  Bayeux  tapestry  as  a  propitious  omen  for 
William  the  Conqueror  (of  England).  In  1456  the  comet,  according  to 
popular  belief,  was  formally  excommunicated  by  Pope  Calixtus  III.  in  a 
bull  directed  mainly  against  the  Turks,  who  were  then  threatening  eastern 
Europe.  It  is  doubtful,  however,  whether  such  a  formal  bull  was  ever  really 
promulgated. 

743.  (2)  Encktfs  Comet.  This  is  interesting  as  the  first  of  the 
short-period  comets,  and  also  as  the  comet  having  the  shortest  known 
time  of  revolution,  —  only  about  three  years  and  a  half.  Encke  first 
detected  its  periodicity  in  1819,  but  it  had  been  frequently  observed 
during  the  preceding  fifty  years,  and  has  been  observed  at  almost 
every  return  since  then.  It  is  usually  visible  only  in  the  telescope, 
though  sometimes,  under  very  favorable  circumstances,  it  can  be 
seen  by  the  naked  eye,  with  a  tail  a  degree  or  two  long.  It  is  often 
irregular  in  form,  and  "  lumpy,"  and  seldom  shows  a  well-defined 
nucleus ;  nor  does  it  exhibit  very  much  that  is  interesting  in  the 
way  of  jets,  envelopes,  and  other  cometary  freaks.  We  have  already 
mentioned  its  remarkable  contraction  in  volume  on  approaching  the 
sun  (Art.  715),  and  the  progressive  shortening  of  its  period,  which 
has  been  ascribed  to  a  resisting  medium  (Art.  710). 


REMARKABLE    COMETS.  457 

744.  (3)   Biela's  Comet.     This  is  also,  or  rather  was,  a  small 
comet  with  a  period  of  6.6  years,  —  the  second  comet  of  short  period 
in  order  of  discovery.     Its  history  is  very  interesting.     It  was  dis- 
covered in  1826  by  Biela,  an  Austrian  officer,  and  its  periodic  char- 
acter was  soon  detected  by  Gambart,  whose  name  is  connected  with 
it  by  many  French  authorities.     Its  orbit  comes  within  a  very  few 
thousand  miles  of  the  earth's  orbit,  the  nearness  varying,  of  course, 
from  time  to  time,  on  account  of  perturbations.     The  approach  is 
often  so  close,  however,  that  if  the  comet  and  the  earth  were  to 
arrive  at  the  nearest  point  at  the   same  time  there  would  be  a 
collision,  and  the  earth  would  pass  through  the  outer  portions  of  the 
comet's  head.    At  the  return  of  the  comet  in  1832,  some  one  started 
the  report  that  such  an  encounter  would  occur,  and  in  consequence 
there  was  something  hardly  short  of  a  panic  in  southern  France  —  the 
first  of  the  since  numerous  "  comet-scares."    At  this  time  the  comet 
passed  the  critical  point  about  a  month  before  the  earth  reached  it, 
so  that  the  two  bodies  were  never  really  within  15000000  miles  of 
each  other. 

745.  At  the  comet's  next  return  in  1839  it  failed  to  be  observed 
on  account  of  its  unfavorable  position  in  the  sky  ;  but  in  1846  it  duly 
reappeared,  and  did  something  very  strange  and  then  unprecedented.1 
It  divided  into  two  !     When  first  seen  on  November  28,  it  presented 
the  ordinary  appearance  of  any  newly  discovered  comet.    On  December 
19  it  had  become  rather  pear-shaped,  and  ten  days  later  it  had 
divided,  the  duplication  being  first  seen  in  New  Haven,  and  soon 
after  at  Washington,  some  days  before  any  European  astronomer 
had  noticed  it. 

The  twin  comets  travelled  along  side  by  side  for  more  than  four  months, 
at  an  almost  unvarying  distance  of  about  160000  miles,  without  showing 
the  least  sign  of  mutual  attraction  or  disturbance  ;  but  internally  both 
comets  were  intensely  active,  each  developing  a  nucleus  very  bright  for  a 
telescopic  comet,  with  a  tail  some  half  a  degree  in  length,  and  showing 
curious  fluctuations  of  light,  which  seemed  as  a  general  rule  to  alternate. 
At  times  the  two  comets  were  connected  by  a  faint  arc  of  light. 

When  next  the  comet  returned  in  August,  1852,  it  was  under  rather 
unfavorable  circumstances  for  observation,  but  the  twins  were  both  seen, 
now  separated  by  about  1  500000  miles,  and  travelling  quietly  in  their 
appointed  orbits.  Neither  of  them  has  ever  been  seen  again,  although  they 
ought  to  have  returned  five  times,  and  more  than  once  under  favorable  con. 
ditions  for  visibility. 

1  See  also  Arts.  740*  and  751.     Also  note  at  end  of  chapter. 


458  COMETS. 

746.  But  the  story  is  not  yet  ended,  though  the  remainder  per- 
haps belongs  more  properly  to  the  next  chapter  of  our  book. 

On  the  night  of  November  27, 1872,  just  as  the  earth  was  passing 
the  old  track  of  the  lost  comet,  she  encountered  a  wonderful  meteoric 
shower.  As  Miss  Clerke  expresses  it,  perhaps  a  little  too  positively, 
"It  became  evident  that  Biela's  comet  was  shedding  over  us  the 
pulverized  products  of  its  disintegration." 

The  same  thing  happened  again  in  November,  1885,  and  1892,  when 
the  earth  once  more  passed  the  comet's  path. 

The  meteors  of  this  so-called  Bielid  swarm,  in  their  motion  through 
the  sky,  all  appear  to  come  from  a  point  in  the  constellation  of 
Andromeda,  and  are  therefore  sometimes  called  the  "Andromedes," 
and  their  motion  is  parallel  to  the  comet's  orbit,  at  the  point  where 
it  intersects  our  own. 

747.  (4)  Donates  Comet  of  1858.     This,  on  the  whole,  was  per- 
haps the  finest  (though  not  the  largest  or  the  most  extraordinary)  of 
the  comets  of  the  present  centurjr,  having  been  very  favorably  situ- 
ated for  observation  in  the  October  sky. 

It  was  discovered  at  Florence  as  a  telescopic  object  on  June  2.  It  did 
not,  however,  become  visible  to  the  naked  eye  until  near  the  end  of  August, 
when  it  began  to  exhibit  the  beautiful  phenomena  which  have  made  it,  so 
to  speak,  the  normal  and  typical  comet.  The  comet  had  an  apparently 
well-defined  nucleus,  which  varied  in  diameter  at  different  times  from  500 
miles  to  3000.  For  several  weeks  the  coma  exhibited  in  unrivalled  perfection 
the  development  and  structure  of  concentric  envelopes.  Its  tail  was  of  the 
second  or  hydrocarbon  type,  with  faint  tangential  streamers  which  belong 
to  the  first  or  hydrogen  type ;  it  had  a  maximum  apparent  length  of  about 
50°,  and  was  some  5°  or  6°  wide  at  the  extremity,  and  its  real  length  was 
about  45000000  miles,  with  a  width  of  10000000.  The  object  was  kept 
under  accurate  observation  for  fully  nine  months,  so  that  its  orbit  is  unusu- 
ally well  determined  as  a  very  long  ellipse,  with  a  periodic  time  of  nearly 
2000  years.  Figs.  197  and  199  show  its  principal  features. 

Our  space  permits  us  to  cite  in  detail  only  one  other  comet :  — 

748.  (5)  The  Great  Comet  0/1882,  which  will  always  be  remem- 
bered, not  only  for  its  beauty,  but  for  the  great  variety  of  unusual 
phenomena  it  presented. 

Discovery  and  Brightness.  The  comet  seems  to  have  been  first  seen 
as  a  naked-eye  object,  at  Auckland,  New  Zealand,  on  September  3. 
By  the  7th  or  8th  it  had  become  somewhat  conspicuous,  and  was 
observed  both  at  Cordova  (South  America)  and  at  the  Cape  of  Good 


REMARKABLE   COMETS.  459 

Hope,  but  was  not  seen  in  the  north,  until  the  day  when  it  passed  its 
perihelion,  September  17.  It  was  then  independently  discovered  by 
Common,  in  England,  in  broad  daylight,  within  2°  of  the  sun ;  and 
the  next  day  it  was  similarly  discovered  by  a  number  of  observers, 
especially  by  Thollon,  at  Nice,  who  observed  its  spectrum  in  full 
sunlight,  and  measured  the  displacement  of  the  sodium  lines  pro- 
duced by  its  motion.  It  was  so  bright  that  there  was  not  the 
slightest  difficulty  in  seeing  it  by  simply  shutting  off  the  sun  with 
the  hand  held  at  arm's  length. 

750.  Member  of  a  Comet  Group.  —  As  has  been  stated  before 
(Art.  705),  its  orbit  —  at  least,  that  portion  of  it  within  the  earth's 
orbit  —  coincides  almost  exactly  with  the  orbits  of  the  comets  of 
1668,  1843,  and  1880.     The  salient  peculiarity  of  these  orbits  lies 
in  the  closeness  of  their  approach  to  the  sun,  the  perihelion  distance 
of  each  of  them  being  less  than  750000  miles,  so  that  they  all 
passed  within  300000  miles  of  the  sun's  surface,  and  with  a  velocity 
which  at  perihelion  exceeded  250  miles  per  second,  and  carried 
them  through  180°  of  their  orbit  in  less  than  three  hours.    And  yet, 
this  passage  through  the  sun's  coronal  regions  did  not  disturb  their 
motion  in  the  least,  as  is  shown  by  the  fact  that  the  orbit  of  the 
comet  of  1882,  deduced  from  the  observations  made  before  the  peri- 
helion passage,  agrees  exactly  with  that  deduced  from  those  made 
after  it.     The  inference  as  to  the  extreme  rarity  of  the  sun's  corona' 
is  obvious.     Only  one  other  comet  —  Newton's  comet  of  1680  —  has 
ever  approached  even  nearly  as  close  to  the  sun  as  the  four  comets 
of  this  group. 

The  comet  continued  visible  until  March,  and  this  long  period  of  observa- 
tion enabled  the  computers  to  determine  the  orbit  with  a  greater  degree  of 
accuracy  than  is  usual.  They  all  agree  in  making  it  a  very  elongated 
ellipse,  with  a  period  ranging  from  650  years  to  840  years,  according  to 
different  computers. 

751.  Telescopic  Features.  —  When  the  comet  first  became  tele- 
scopically  observable  in  the  morning  sky  it  presented  very  nearly  the 
normal  appearance.     The  nucleus  was  sensibly  circular,  and  there 
were  a  number  of  clearly  developed,  concentric  envelopes  in  the 
head;  the  dark,  shadow-like  stripe  behind  the  nucleus  was  also  well 
marked.     In  a  few  days  the  nucleus  became  elongated,  and  finally 
stretched  out  into  a  lengthened,  luminous  streak  some  50000  miles 
in  extent,  upon  which  there  were  six  or  eight  star-like  knots  of  con- 
densation.    The  largest  and  brightest  of  these  knots  was  the  third 


460  COMETS. 

from  the  forward  end  of  the  line,  and  was  some  5000  miles  in 
diameter.     This  "string  of  pearls"  continued  to  lengthen  as  long  as 


Oct.  9.  Oct.  15. 

FIG.  206.  —  The  Head  of  the  Great  Comet  of  1882. 

the  comet  was  visible,  until  at  last  the  length  exceeded  100000  miles. 
The  engraving  (Fig.  206)  represents  the  telescopic  appearance  at 
Princeton  on  October  9  and  15. 

752.  Tail.  —  The  comet  was  so  situated  that  its  tail  was  not  seen 
to  the  best  advantage,  being  directed  nearly  away  from  the  earth, 
and  never  having  an  apparent  length  much  exceeding  35°.  The 
actual  length  of  the  tail,  however,  at  one  time  exceeded  100000000 
miles,  —  more  than  the  distance  of  the  earth  from  the  sun.  It  was 
of  the  second  or  hydrocarbon  type. 

A  unique  and  so  far  unexplained  phenomenon  was  a  faint,  straight- 
edged  beam  of  light,  or  "sheath"  that  accompanied  the  comet, 
enveloping  the  head  and  projecting  three  or  four  degrees  in  front 
of  it,  as  shown  in  the  figure  (Fig.  207).  Besides  this,  at  different 
times,  three  or  four  irregular  shreds  of  cometary  matter  were 
detected  by  Schmidt,  of  Athens,  and  other  observers,  accompany- 
ing the  comet  at  a  distance  of  three  or  four  degrees  when  first 
seen,  but  gradually  receding  from  it,  and  at  the  same  time  growing 
fainter. 


FIG.  208.  —  Rordame's  Comet,  1893. 
(Photographed  by  Hussey.) 


TELESCOPIC    FEATURES. 


461 


FIG.  207.  —The  "  Sheath,"  and  the  Attendants  of  the  Comet  of  1882. 

752*  Photography  of  Comets.  —  Mr.  Gill  at  the  Cape  of  Good 
Hope  obtained  a  number  of  fairly  good  photographs  of  the  comet 
of  1882,  and  since  then  the  art  has  so  improved  that  it  is  now 
possible  to  bring  out  with  the  camera  peculiarities  and  details  which 
are  quite  invisible  to  the  eye  even  in  powerful  telescopes.  This  is 
especially  the  case  with  the  comet's  tail.  Fig.  208  is  from  a  photo- 
graph of  Rordame's  comet  of  1893,  for  which  we  are  indebted  to  the 
kindness  of  Professor  Holden,  of  the  Lick  Observatory.  Because 
the  camera  (strapped  to  a  telescope  tube)  was  of  course  kept  pointed 
at  the  head  of  the  comet,  which  was  moving  rapidly,  the  images  of 
the  stars  in  the  field  of  view  during  the  hour's  exposure  are  drawn 
out  into  parallel  streaks,  the  little  irregularities  being  due  to  faults 
of  the  clock-work  and  vibrations  of  the  telescope.  The  knots  and 
streamers  which  characterize  the  comet's  tail  were  none  of  them 
visible  in  the  telescope,  and  are  not  the  same  shown  upon  plates, 
taken  the  day  before  and  the  day  after.  Other  plates,  made  the 
same  evening  a  few  hours  earlier  and  later,  indicate  that  the  " knots" 
were  swiftly  receding  from  the  comet's  head  at  a  rate  exceeding 
150000  miles  an  hour. 

In  1892  Barnard  discovered  a  small  comet,  by  the  streak  it  left  upon  one 
of  his  star-plates. 


462  COMETS. 

We  close  the  chapter  with  a  few  remarks  upon  a  subject  which 
has  been  much  discussed. 

753.  The  Earth's  Danger  from  Comets.  —  It  has  been  supposed 
that  comets  might  do  us  harm  in  two  ways,  —  either  by  actually 
striking  the  earth,  or  by  falling  into  the  sun,  and  thus  producing 
such  an  increase  of  solar  heat  as  to  burn  us  up. 

As  regards  the  possibility  of  a  collision  with  a  comet,  it  is  to  be 
admitted  that  such  an  event  is  possible.  In  fact,  if  the  earth  lasts 
long  enough,  it  is  practically  sure  to  happen ;  for  there  are  several 
comets'  orbits  which  pass  nearer  to  the  earth's  orbit  than  the  semi- 
diameter  of  the  comet's  head,  and  at  some  time  the  earth  and  comet 
will  certainly  come  together.  Such  encounters  will,  however,  be 
very  rare.  If  we  accept  the  estimate  of  Babinet,  they  will  occur 
about  once  in  15  000000  years  in  the  long  run. 

As  to  the  consequence  of  such  a  collision  it  is  impossible  to  speak 
with  confidence,  for  want  of  sure  knowledge  of  the  state  of  aggrega- 
tion of  the  matter  composing  a  comet.  If  the  theory  presented  in 
this  chapter  is  true,  everything  depends  on  the  size  of  the  separate 
solid  particles  which  form  the  main  portion  of  the  comet's  mass. 
If  they  weigh  tons,  the  bombardment  experienced  by  the  earth  when 
struck  by  a  comet  would  be  a  very  serious  matter ;  if,  as  seems  more 
likely,  they  are  for  the  most  part  smaller  than  pin-heads,  the  result 
would  be  simply  a  meteoric  shower. 

A  danger  of  a  different  sort  has  been  suggested,  that  if  a  comet  were  to 
strike  the  earth  our  atmosphere  would  be  poisoned  by  the  mixture  with 
the  gaseous  components  of  the  comet.  Here  again  the  probability  is  that 
on  account  of  the  low  density  of  the  cometary  matter  no  sufficient  amount 
of  deleterious  vapors  would  remain  in  the  air  to  do  any  mischief  at  the 
earth's  surface. 

754.  Effect  of  the  Fall  of  a  Comet  into  the  Sun. — As  regards  the 
effect  of  the  fall  of  a  comet  into  the  sun,  it  may  be  stated  that,  except 
in  the  case  of  Encke's  comet,  there  is  no  evidence  of  any  action  going 
on  that  would  cause  a  now  existing  periodic  comet  to  strike  the  sun's 
surface ;  it  is,  however,  undoubtedly  possible  that  a  comet  may  enter 
the  system  from  without,  so  accurately  aimed  that  it  will  hit  the  sun. 

But,  if  a  comet  actually  strikes  the  sun,  it  is  not  likely  that  the 
least  harm  will  be  done.  If  a  comet  having  a  mass  equal  to  T^VoT?  of 
the  earth's  mass  were  to  strike  the  sun's  surface  with  the  parabolic 
velocity  of  nearly  400  miles  a  second,  it  would  generate  about  as 
much  heat  as  the  sun  radiates  in  eight  or  nine  hours.  If  this  were 


EFFECT  OF  THE  FALL  OF  A  COMET  INTO  THE  SUN.          463 

all  instantly  effective  in  producing  increased  radiation  at  the  sun's 
surface  (increasing  it,  say,  eightfold,  for  even  a  single  hour),  mis- 
chief would  follow,  of  course.  But  it  is  almost  certain  that  nothing 
of  the  sort  would  happen.  The  cometary  particles  would  pierce  the 
photosphere,  and  liberate  their  heat  mostly  below  the  solar  surface, 
simply  expanding,  by  some  slight  amount,  the  sun's  diameter,  and 
so  adding  to  its  store  of  potential  energy  about  as  much  as  it 
ordinarily  expends  in  a  few  hours.  There  might,  and  very  likely 
would,  be  a  flash  of  some  kind  at  the  solar  surface,  as  the  shower  of 
cometary  particles  struck  it,  but  probably  nothing  that  the  astron- 
omer would  not  take  delight  in  watching. 


NOTE. 

Callandreau  has  very  recently  published  an  important  paper  upon  the 
disintegration  of  comets  by  the  action  of  the  sun  and  the  planet  Jupiter, 
showing  that  the  limiting  distance  at  which  such  an  effect  is  possible  is 
quite  considerable,  and  that  the  breaking  up  of  a  comet  ought  not  to  be 
very  unusual.  He  suggests  that  many  of  the  "  comet  groups  "  may  have 
originated  in  this  way,  arid  that  the  number  of  the  comets  in  Jupiter's 
family  has  probably  thus  been  greatly  increased.  The  difficulty  referred 
to  in  the  last  sentence  of  Art.  740  respecting  the  "  capture  theory  "  is  thus 
very  much  relieved.  (March,  1898.) 


EXERCISES  ON  CHAPTER  XVIII. 
\j 

1.  What  would  be  the  mean  density,  compared  with  air,  of  the  spherical 
head  of  a  comet  a  hundred  thousand  miles  in  diameter,  and  having  a  mass 
one  hundred-thousandth  that  of  the  earth ;  assuming  the  density  of  the 
earth  as  5.55  times  that  of  water,  and  the  density  of  water  as  773  times 

that  of  air?  A          Ai  , 

Ans.    About  ^y^j^y 

*    2.    What  would  be  the  diameter  of  such  a  comet  if  compressed  to  a 
density  the  same  as  that  of  the  earth?  ,         -,*-,       •-, 

3.  Can  the  dimensions  of  a  comet's  tail  be  determined  with  much 
accuracy?  If  not,  why  not? 


464  EXERCISES. 

4.  How  can  it  happen  that  comets  whose  orbits  nearly  coincide  within  a 
distance  of  a  hundred  million  miles  from  the  sun  may  have  periods  differing 
by  hundreds  of  years?    For  example  the  comets  of  1880  and  1882,  of  which 
the  first  has  a  computed  period  of  only  33  years,  and  the  other,  of  more 
than  600. 

5.  In  the  case  of  two  cometary  orbits  very  nearly  parabolic,  and  having 
the  same  very  small  perihelion  distance,  how  would  the  ratio  of  their  major- 
axes  be_affected^by  a  small  difference  in  their  perihelion  velocities?     (See 
Art.  429,  remembering  that,  as  the  orbits  are  nearly  parabolic,  V2  must  be 
very  nearly  equal  to  £72  when  the  comets  pass  perihelion.) 

6.  If  the  repulsive  force  of  the  sun  upon  a  particle  of  a  comet's  tail  were 
just  equal  to  the  gravitational  attraction  (Art.  728)  what  would  be  the 
path  of  that  particle?  ^    A  gtraight  ^ 

7.  If  the  repulsive  force  exceeded   the    gravitational   attraction  what 
would  be  the  nature  of  the  path  ? 

Ans.    An  hyperbola  convex  towards  the  sun,  _  and  with  the  sun  in  the 
external  focus. 

8.  What  would  be  the  path  if  the  repulsive  force  were  only  very  small 
as  compared  with  the  gravitational  attraction? 

Ans.    An  orbit  of  slightly  greater  eccentricity  than  that  of  the  comet 
itself. 

9.  Will  a  given  comet  (say  Encke's)  have  precisely  the  same  orbit  on 
successive  returns? 

10.  Why  can  we  not  infer  with  certainty  that  two  comets  which  have 
orbits  practically  identical  are  themselves  identical? 

11.  Can  we  from  spectroscopic  observations  of  a  comet  infer  the  relative 
proportions  of  the  luminous  and  non-luminous  substances  present  in  the 
comet? 

12.  Is  it  probable  that  a  comet  can  continue  permanently  in  the  solar 
system  as  a  comet  ?     If  not,  why  not,  and  what  will  become  of  it  ? 


METEORS.  465 


CHAPTER    XIX. 

METEORS   AND    SHOOTING   STARS. 

755.  Meteors.  —  Occasionally  bodies  fall  on  the  earth  from  the 
sky,  —  masses  of  stone  or  iron  which  sometimes  weigh  several  tons. 
During  its  flight  through  the  sky  such  a  body  is  called  a  meteor,  and 
the  pieces  which  fall  from  it  are  called  meteorites,  or  aerolites  (air- 
stones),  or  uranoliths  (heaven-stones),  or  simply  meteoric  stones. 

756.  Circumstances  of  their  Fall.  —  The   circumstances   which 
attend  the  fall  of  a  meteorite  are  in  most  cases  substantially  as 
follows.     If  it  occurs  at  night  a  ball  of  fire  is  seen,  which  moves 
with  an  apparent  speed  depending  both  on  its  real  velocity  and  on 
uhe  observer's  position.     If  the  body  is  coming  "head  on,"  so  to 
speak,  the  motion  will  be  comparatively  slow  ;  so  also  if  it  is  very 
distant.     The  fire-ball  is  generally  followed  by  a  luminous  train, 
which  marks  out  the  path  of  the  body,  and  often  continues  visible 
for  a  long  time  after  the  meteor  itself  has  disappeared.    The  motion 
is  seldom  exactly  straight,  but  is  more  or  less  irregular,  owing  to  the 
resistance  of  the  air  ;  and  every  here  and  there  along  its  path  the 
meteor  seems  to  throw  off  fragments,  and  to  change  its  course  more 
or  less   abruptly.     If  the   observer  is  near  enough,  the  flight  is 
accompanied   by   a  heavy,  continuous  roar,  accentuated  by  sharp 
detonations  which  accompany  the  visible  explosions  by  which  frag- 
ments are  burst  off  from  the  principal  body.    The  noise  is  sometimes 
tremendous,  and  heard  for  distances  of  forty  or  fifty  miles,  but  since 
sound  travels  only  about  1100  feet  a  second  the  explosions,  if  distant, 
are  heard  after  a  considerable  interval,  —  often  several  minutes. 

If  the  fall  occurs  by  day  white  clouds  take  the  place  of  the  fire- 
ball and  the  train. 

757.  The  Aerolites  Themselves.  —  The  mass  that  falls  is  some- 
times a  single  piece,  but  more  usually  there  are  many  separate  frag- 
inentSj  as  in  the  case  of  the   Spanish  meteors  of  January,  1896. 
Sometimes  they  number  thousands,  as  in  the  L'Aigle  meteors  of 
1803 ;  then,  naturally,  the  stones  are  mostly  small,  and  sometimes 
they  are  mere  grains   of  sand.     Nearly  all   the  aerolites  that  are 
actually  seen  to  fall,  and  are  found  at  the  time,  are  masses  of  stone  ; 


466  METEORS. 

but  a  very  few,  perhaps  three  or  four  per  cent  of  the  whole  number, 
consist  of  nearly  pure  iron,  more  or  less  alloyed  Avith  nickel.  There 
are  also  a  good  many  cases  of  uranoliths,  which  are  mainly  stony, 
but  have  a  considerable  portion  of  iron  disseminated  through  the 
mass  in  grains  and  globules  ;  and  nearly  all  the  stony  uranoliths 
contain  as  much  as  twenty  or  thirty  per  cent  of  iron  in  the  form  of 
sulphides  or  analogous  compounds. 

758.  The  only  iron  meteors  which  have  been  actually  seen  to  fall  so  far, 
and  are  represented  by  specimens  in  our  museums,  are  the  following :  — 

(1)  Agram,  Bohemia, 1751. 

(2)  Dickson  Co.,  Tennessee, 1835. 

(3)  Braunau,  Bohemia, 1847. 

(4)  Tabarz,  Saxony,  ........  1854. 

(5)  Nejed,  Arabia, 1865. 

(6)  Nedagollah,  India, 1870. 

(7)  Maysville,  California, 1873. 

(8)  Rowton,  Shropshire,  England,     .     .     .  1876. 

(9)  Emmett  Co.,  Iowa, 1879. 

(10)  Mazapil,  Mexico, 1885. 

(11)  Johnson  Co.,  Arkansas, 1886. 

The  Emmett  County  iron  was  mostly  in  small  fragments,  and  along  with 
them  there  were  many  large  stones  with  quantities  of  iron  included.  The 
separate  fragments  of  pure  iron  which  reached  the  earth  probably  came  by 
the  breaking  up  of  the  stony  masses. 

Besides  these  iron  meteors  which  have  been  seen  to  fall,  our  cabinets 
contain  a  very  large  number  of  so-called  meteoric  irons ;  i.e.,  masses  of  iron 
found  under  such  circumstances  that  they  cannot  easily  be  accounted  for  in 
any  way  except  by  supposing  them  to  be  of  meteoric  origin. 

759.  The   number  of   meteorites   which   have  fallen  since   1800   and 
been   gathered  into  our  cabinets  *  is  about  275.      The  most  remarkable 
falls  in  the  United  States  have  been  the  six  following:  namely,  Weston, 
Connecticut,  1807 ;    Bishopsville,  So.  Carolina,  1843  ;    Cabarrus  Co.,   No. . 
Carolina,  1849  ;  New  Concord,  Ohio,  1860  ;  Amana,  Iowa,  1875  ;  and  Em- 
mett Co.,  Iowa,  1879.     In  the  first  case  and  the  three  last,  several  hundred 
fragments  fell  at  the  same  time,  ranging  in  size  from  five  hundred  pounds 
to  half  an  ounce. 

760.  Twenty-five  of  the  chemical  elements,  including  helium,  have  been 
found   in  meteors,   and   not  a  single  new  one.      The  minerals  of  which 

1  In  this  country  the  cabinets  of  Amherst  College  and  Harvard  and  Yale 
Universities  are  especially  rich  in  meteorites.  The  finest  collection  in  the  world, 
however,  is  that  at  Vienna.  The  collection  of  the  British  Museum  is  also  note- 
worthy, as  well  as  that  at  Paris. 


THE   AEROLITES   THEMSELVES. 


467 


meteorites  are  composed  present  a  great  resemblance  to  terrestrial  minerals 
of  volcanic  origin,  but  many  of 
them  are  peculiar,  and  found  in 
meteors  only.  (The  study  of 
these  meteoric  minerals  is  a  very 
curious  and  important  branch 
of  mineralogy,  though  natu- 
rally it  has  not  many  votaries.) 
The  occasional  presence  of  car- 
bon is  to  be  specially  noted,  and 
in  a  meteor  which  fell  in  Russia 
the  carbon  appeared  to  be  in 
a  crystalline  form,  identical 
with  the  black  diamond,  though 
in  exceedingly  minute  particles. 
Fig.  209  is  from  a  photograph 
of  a  fragment  of  one  of  the 
meteoric  stones  which  fell  at 
Amana,  Iowa,  in  1875.  The 
picture  is  taken  by  the  permis- 
sion of  the  publishers  from 
Professor  Langley's  "  New  As- 
tronomy," where  the  body  is 
designated  as  "  part  of  a  comet." 


FIG.  209. 
Fragment  of  one  of  the  Amana  Meteoric  Stones. 


761.  The  Crust.  —  The  most  characteristic  external  feature  of  an 
aerolite  is  the  thin,  black  crust  that  covers  it,  usually,  but  not  always, 
glossy  like  a  varnish.     It  is  formed  by  the  fusion  of  the  surface  in 
the  meteor's  swift  motion  through  the  air,  and  in  some  cases  pene- 
trates deeply  into  the  mass  of  the  meteor  through  fissures  and  veins. 
It  is  largely  composed  of  oxide  of  iron,  and  is  always  strongly 
magnetic.     The  crusted  surface  usually  exhibits  pits  and  hollows, 
such  as  would  be  produced  by  thrusting  the  thumb  into  a  mass  of 
putty.     These  cavities  are  explained  by  the  burning  out  of  certain 
more  fusible  substances  during  the  meteor's  flight. 

762.  Magnitude. —  Of  the  meteors  actually  seen  to  fall  the  largest 
pieces  found  thus  far  weigh  about  500  pounds,  though  the  whole  mass 
of  the  body  when  it  first  entered  the  atmosphere  has  sometimes  been 
much  larger,  perhaps,  in  a  few  cases,  amounting  to  two  or  three  tons.1 

1  Some  of  the  masses  of  iron  supposed  to  be  of  meteoric  origin,  but  not  actually 
seen  to  fall,  are  very  much  larger.  The  iron  mass  from  Otumpa  in  Mexico  is  said 
to  weigh  fully  sixteen  tons.  As  regards  some  of  these  hypothetical  meteorites, 


468  METEORS. 

As  seen  from  a  distance  of  many  miles,  the  meteoric  fire-ball  some- 
times appears  to  have  a  diameter  as  large  as  the  moon,  which  would 
indicate  a  real  diameter  of  several  hundred  feet.  The  great  apparent 
size,  however,  is  an  illusion,  partly  due  to  irradiation,  and  partly, 
undoubtedly,  to  the  fact  that  the  meteor  itself  is  surrounded  by  an 
extensive  envelope  of  heated  air  and  smoke  which  becomes  luminous 
throughout.  Probably  no  single  meteor  ever  yet  investigated  was  a 
solid  mass  as  large  as  ten  feet  in  diameter. 

763.  Path. — When  a  meteor  has  been  observed  by  a  number  of 
persons  at  different  points,  who  have  noted  any  data  which  will  give 
its  altitude  and  bearing  at  identified  moments,  the  path  can  be  com- 
puted.    Observations  from  a  single  point  are  worthless  for  the  pur- 
pose, since  they  can  give  no  information  as  to  the  meteor's  distance. 

The  meteor  is  generally  first  seen  at  an  altitude  of  between  eighty 
and  100  miles,  and  disappears  at  an  altitude  of  between  five  and  ten 
miles.  The  length  of  the  path  may  be  anywhere  from  50  miles  to 
500,  according  to  its  inclination  to  the  earth's  surface.  The  velocity 
is  rather  difficult  to  ascertain,  but  is  found  to  range  from  ten  to  fort}' 
miles  per  second  at  the  moment  when  the  meteor  first  becomes  visible, 
and  diminishes  to  one  or  two  miles  per  second,  at  the  time  when  it 
disappears.  The  average  velocity  with  which  meteors  enter  the 
atmosphere  appears  not  to  vary  much  from  the  "  parabolic  velocity  " 
of  twenty-six  miles  per  second,  due  to  the  sun's  attraction  at  the 
earth's  distance  —  a  fact  which,  of  course,  indicates  that  these  bodies, 
whatever  their  origin  may  be,  are  now  moving  in  space,  like  the  comets, 
under  the  sun's  attraction. 

With  possibly  a  very  few  exceptions  in  cases  where  the  meteor  glances, 
so  to  speak,  on  the  earth's  atmosphere,  like  a  skipping-stone  on  water,  a 
body  which  has  once  entered  the  air  is  sure  to  be  brought  to  the  ground : 
it  is  hardly  possible  that  one  meteor  in  a  million  should  escape  after  becom- 
ing involved  in  the  atmosphere.  We  mention  this  especially,  because  some 
authorities  erroneously  speak  of  it  as  a  usual  thing  for  the  meteor  to  keep 
on  its  course,  and  leave  the  earth,  after  throwing  off  a  few  fragments. 

764.  Observation   of  Meteors.  —  The   object  of   the  observation 
should  be  to  obtain  accurate  estimates  of  the  altitude  and  azimuth  of 
the  body  at  moments  which  can  be  identified.     At  night  this  is  best 

however,  their  meteoric  origin  is  extremely  questionable  ;  such,  for  instance,  is 
the  case  with  the  enormous  masses  of  iron,  one  of  them  weighing  more  than 
seventy  tons,  brought  from  the  Greenland  coast  by  Nordenskiold  and  Peary. 


THEIR    PATHS.  469 

done  by  noting  the  position  of  the  meteor  with  reference  to  neighbor- 
ing stars  at  the  moments  of  its  appearance  and  disappearance,  or  of 
the  intervening  explosions.  In  the  daytime  it  can  often  be  done  by 
noting  the  position  of  the  object  with  reference  to  trees  or  buildings. 
The  observer  should  then  mark  the  exact  position  where  he  is  standing, 
so  that  by  going  there  afterwards  with  proper  instruments  he  can 
determine  the  data  desired. 

Of  course,  all  such  measurements  must  be  given  in  angular  units.  To 
speak  of  a  meteor  as  having  an  altitude  of  twenty  feet,  and  pursuing  a  path 
100  feet  long,  is  meaningless,  unless  the  size  of  the  "  foot  "  is  somehow 
denned. 

The  determination  of  the  meteor's  velocity  is  more  difficult,  as  it 
is  seldom  possible  to  look  at  a  watch-face  quickly  enough,  even  in 
the  daytime.  The  usual  course  is  for  the  observer  to  repeat  some 
familiar  piece  of  doggerel  as  rapidly  as  possible,  beginning  when  the 
object  first  becomes  visible  and  stopping  when  it  explodes  or  disap- 
pears, noting  also  the  precise  syllable  where  he  stops.  By  repeating 
the  same  sentence  over  again  before  a  clock  it  is  possible  to  deter- 
mine within  a  few  tenths  .of  a  second  the  time  occupied  by  the 
meteor's  flight. 

765.   Explanation  of  the  Heat  and  Light  of  a  Meteor.  —  These 

are  due  simply  to  the  destruction  of  the  body's  velocity  ;  its  kinetic 
mass-energy  of  motion  is  transformed  into  heat  by  the  friction  of  the 
air.  If  a  moving  body  whose  mass  is  M  kilograms,  and  its  velocity 
V  metres  per  second,  is  stopped,  the  number  of  calories  of  heat 
developed  is  given  by  the  equation 


The  quantity  of  heat  evolved  in  bringing  to  rest  a  body  which  has  a 
velocity  of  forty-two  kilometres,  or  twenty-six  miles  a  second,  is  enor- 
mous, vastly  more  than  sufficient  to  fuse  it  even  if  it  were  made  of 
the  most  refractory  material,  and  hundreds  of  times  more  than  would 
be  produced  by  its  combustion  in  oxygen  if  it  were  a  mass  of  coal. 

This  heat  is  developed  all  along  the  meteor's  course,  and  mostly  just 
upon  its  surface.  As  Sir  William  Thomson  has  shown,  the  thermal 
effect  of  the  rush  through  the  air  is  the  same  as  if  the  meteor  were 
immersed  in  a  How-pipe  flame  having  a  temperature  of  many  thou- 
sand degrees  ;  and  it  is  to  be  noted  that  this  temperature  is  indepen- 


470  METEOES. 

dent  of  the  density  of  the  air  through  which  the  meteor  may  be  passing. 
The  quantity  of  heat  developed  in  a  given  time  is  greater,  of  course, 
where  the  air  is  dense  j  but  the  temperature  produced  in  the  air  itself, 
at  the  surface  where  it  rubs  against  the  moving  body,  is  the  same 
whether  the  gas  be  dense  or  rare. 

When  a  moving  body  has  a  velocity  of  about  1500  metres  per  second, 
the  virtual  temperature  of  the  surrounding  air  is  about  that  of  red  heat ;  i.e., 
the  body  becomes  heated  as  fast  as  it  would  if  it  were  at  rest  and  the  air 
about  it  were  heated  to  that  temperature.1  When  the  velocity  reaches  twenty 
or  thirty  miles  per  second,  it  is  acted  upon  as  if  the  surrounding  gas  were 
heated  to  the  liveliest  incandescence  at  a  temperature  of  several  thousand 
degrees.  The  surface  is  fused,  and  the  liquefied  portion  is  continually 
swept  off  by  the  rush  of  the  air,  condensing  as  it  cools  into  the  luminous 
powder  that  forms  the  train.  The  fused  surface  itself  is  continually 
renewed  until  the  velocity  falls  below  two  miles  a  second  or  thereabouts, 
when  it  solidifies  and  forms  the  characteristic  crust.  As  a  general  rule, 
therefore,  the  fragments  are  hot  if  found  soon  after  their  fall ;  but  if  the 
stone  is  a  large  one  and  falls  nearly  vertically,  so  as  to  have  but  a  short  path 
through  the  air,  the  heating  effect  will  be  mainly  confined  to  its  surface ;  and 
owing  to  the  low  conducting  power  of  stone,  the  centre  may  still  remain 
intensely  cold,  retaining  nearly  the  temperature  which  it  had  in  interplanet- 
ary space.  It  is  recorded  that  one  of  the  large  fragments  of  the  Dhurmsala 
(India)  meteorite,  which  fell  in  1860,  was  found  in  moist  earth  half  an  hour 
or  so  after  the  fall,  coated  with  ice. 

766,  Train.  —  One  unexplained  feature  of  meteoric  trains  de- 
serves notice.     They  often  remain  luminous  for  a  long  time,  some- 
times as   much  as  half  an  hour,  and  are  carried  by  the  wind  like 
clouds.    It  is  impossible  to  suppose  that  such  a  cloud  of  dust  remains 
incandescent  from  heat  for  so  long  a  time  in  the  cold  upper  regions  of 
the  atmosphere;  and  the  question  of  its  enduring  luminosity  or  phos- 
phorescence is  an  interesting  and  puzzling  one. 

767.  Origin.  —  We  may  at  once  dismiss  the  theories  which  make 
meteors  to  be  the  immediate  product  of  volcanic  eruption  on  the 
earth  or  on  the  moon.     They  come  to  us  for  the  most  part,  as  has 
been  said,  from  the  depths  of  space,  with  the  velocity  of  planets  and 

1  This  is  because  the  gaseous  molecules  strike  the  surface  of  the  meteor  as  if 
it  were  at  rest,  and  the  molecules  themselves  were  moving  with  speed  corre- 
spondingly increased.  According  to  the  "kinetic  theory"  of  gases  the  "tem- 
perature" of  a  gas  depends  entirely  upon  the  "mean-square  velocity"  of  its 
molecules. 


TRAIN.  471 

comets,  and  there  is  no  certain  reason  for  assuming  that  they  origi- 
nated in  any  manner  different  from  the  larger  heavenly  bodies. 

At  the  same  time,  many  of  them  so  closely  resemble  each  other  as  almost 
to  compel  the  idea  of  some  common  source  ;  and  though  lunar  volcanoes  are 
now  extinct,  and  no  terrestrial  volcano,  not  even  Krakatao,  is  now  competent 
to  send  off  its  ejected  missiles  through*  the  earth's  atmosphere  into  space,  it 
is  not  certain  that  this  was  always  so.  Some  still  maintain  that  these  bodies 
may  be  fragments  which  were  shot  off  millions  of  years  ago  when  the  moon's 
volcanoes  were  in  full  vigor  and  the  earth  was  young.  Since  then,  according 
to  this  view,  these  masses  have  been  travelling  around  the  sun  in  long 
ellipses  which  intersect  the  orbit  of  the  earth,  until  at  last  they  happen  to 
come  along  at  the  right  time  and  encounter  her  atmosphere. 

As  to  the  iron  meteors,  some  maintain  that  they  come  to  us  from  the  sun 
or  some  other  star,  basing  the  opinion  upon  the  remarkable  fact  that  these 
meteoric  irons  are  generally  full  of  "  occluded  "  hydrogen,  helium  and  carbon 
oxides,  which  can  be  extracted  by  proper  methods.  They  argue  that  the  iron 
could  have  absorbed  these  gases  only  while  immersed  in  a  hot  dense  atmos- 
phere containing  them,  —  a  condition  existing,  so  far  as  known,  only  on  the 
sun  and  stars.  There  is  no  doubt  of  the  sun's  ability  to  project  masses 
to  planetary  distances,  as  shown  in  the  case  of  many  eruptive  prominences ; 
and  it  is  not  unreasonable  to  suppose  that  other  suns  can  do  the  same. 

However  these  bodies  originated,  it  is  quite  certain  that  before  they 
reach  the  earth  they  have  been  moving  independently  in  space  for  a 
long  time,  just  as  planets  and  comets  do.  But  a  recent  important 
research  by  the  late  Professor  Newton  has  shown  that  more  than  90 
per  cent  of  some  200  aerolites,  for  the  approximate  determination  of 
whose  paths  we  have  the  data,  were  moving  before  their  fall  in  orbits, 
not  parabolic,  but  analogous  to  those  of  the  short-period  comets; 
and  direct,  not  retrograde. 

768.  Detonating  Meteors,  or  "Bolides,"  of  which  Fragments  are 
not  known  to  reach  the  Earth.  —  Some  writers  discriminate  between 
these  meteors  and  aerolites,  but  the  distinction  does  not  seem  to  be 
well  founded.      The  phenomena  appear  to  be  precisely  the  same, 
except  that  in  the  one  case  the  fragments  are  actually  found,  and  in 
the  other  they  fall  into  the  sea,  the  forest,  or  the  desert ;  or  some- 
times when  the  path  is  nearly  horizontal,  and  therefore  long,  they 
may  be  consumed  and  dissipated  in  the  dust  and  vapor  of  the  train, 
without  reaching  the  earth's  surface  at  all,  except  ultimately  as 
impalpable  dust. 

769.  Number.  —  As  to  the  number  of  aerolites  which  strike  the 
earth,  it  is  difficult  to  make  a  trustworthy  estimate.      Since  the 


472  SHOOTING    STARS. 

beginning  of  the  century,  at  least  two  or  three  have  been  seen  to 
fall  every  year,  and  have  been  added  to  our  cabinets  (see  Art. 
759),  —  in  some  years  as  many  as  half  a  dozen.  This,  of  course, 
implies  a  vastly  greater  number  which  are  not  seen,  or  are  not 
found.  Schreibers,  some  years  ago,  estimated  the  number  as  high 
as  700  a  year,  and  Eeichenbach  sets  it  still  higher  —  not  less  than 
3000  or  4000. 

SHOOTING  STARS. 

770.  A  few  minutes7  watching  on  any  clear,  moonless  night  will 
be  sure  to  reveal  one  or  more  of  the  swiftly  moving,  evanescent 
points  of  light  that  are  known  as  "shooting  stars."     No  sound  is 
ever  heard  from  them,  nor  (with  a  single  exception  to  be  mentioned 
further  on)  has  anything  ever  been  known  to  reach  the  earth's  sur- 
face from  them,  not  even  when  the  sky  was  "as  full  of  them  as  of 
snow-flakes/'  as  sometimes  has  happened  in  a  great  meteoric  shower. 
For  this  reason  it  is  perhaps  justifiable  to  allow  the  old  distinction 
to  remain  between  them  and  the  bodies  we  have  been  discussing,  at 
least  provisionally.     The  difference  may  be,  and  according  to  opinion 
at  present  prevalent  very  probably  is,  merely  one  of  size,  like  that 
between  boulders  and  grains  of  sand.     Still  there  are  some  reasons 
for  supposing  that  there  is  also  a  difference  of  constitution,  —  that 
while  the  aerolite  is  a  solid,  compact  mass,  the  shooting  star  is  a 
little  cloud  of  dust  and  intermingled  gas,  like  a  puff  of  smoke. 

771.  Numbers.  —  The  number  of  these  bodies  is  very  great.     A 
single  watcher  sees  on  the  average  from  four  to  eight  hourly.     If 
observers  enough  are  employed  to  guard  the  whole  sky,  so  that 
none  can  escape  unnoticed,  the  visible  number  becomes  from  thirty 
to  sixty  an  hour.     Since  ordinarily  only  those  are  seen  which  are 
within  two  or  three  hundred  miles  of  the  observer,  the  estimated 
total  daily  number  of  those  which  enter  the  earth's  atmosphere,  and 
are  large  enough  to  be  visible  to  the  naked  eye,  rises  into  the  mil- 
lions.   Professor  Newton  sets  it  at  from  15  000000  to  20  000000,  the 
average  distance  between  them  being  about  250  miles. 

The  number  too  small  to  be  seen  by  the  naked  eye  is  still  larger.  One 
hardly  ever  works  many  hours  with  a  telescope  carrying  a  low  power,  and 
having  a  field  of  view  as  large  as  15'  in  diameter,  without  seeing  several 
of  them  flash  across  the  field.  In  a  few  instances  observers  have  reported 
dark  meteors  crossing  the  moon's  disc  while  they  were  watching  it.  There 
may  be  some  question,  however,  as  to  the  real  nature  of  the  objects  seen  in 
such  a  case.  Birds  (?). 


SHOOTING    STABS.  473 

772.  Comparative  Number  in  Morning  and  Evening. — The  hourly 
number  about  six  o'clock  in  the  morning  is  fully  double  the  hourly 
number  in  the  evening.     The  obvious  reason  is  simply  that  in  the 
morning  we  are  on  the  front  of  the,  earth,  as  regards  its  orbital 
motion,  while  in  the  evening  we  are  in  the  rear ;  in  the  evening  we 
only  see  such  meteors  as  overtake  us  ;  in  the  morning  we  see  all  that 
we  either  meet  or  overtake.1     If  they  are  really  moving  in  all  direc- 
tions alike,  with  the  parabolic  velocity  corresponding  to  the  earth's 
distance  from  the  sun  (twenty-six  miles  per  second),  theory  indicates 
that  the  relative  hourly  numbers  for  morning  and  evening  ought  to 
be  in  just  the  observed  proportion. 

773.  Brightness.  —  For  the  most  part  these  bodies  are  much  like 
the  stars  in  brightness,  —  a  few  are  as  brilliant  as  Venus  or  Jupiter ; 
more  are  like  stars  of  the  first  magnitude ;  and  the  majority  are  like 
the  smaller  stars.     The  bright  ones  not  unfrequently  show  trains 
which  sometimes  last  from  five  to  ten  minutes,  when  they  are  folded 
up  and  wafted  away  by  the  winds  of  the  upper  air.2 

774.  Elevation,  Path,  and  Velocity.  —  By  observations  made  by 
two  or  more  observers  thirty  or  forty  miles  apart,  it  is  possible  to 
determine  the  height,  path,  and  velocity  of  these  bodies.    It  is  found 
as  the  result  of  a  great  number  of  such  observations  that  they  first 
appear  at  an  average  elevation  of  about  seventy-four  miles,  and  dis- 
appear at  an  average  height  of  about  fifty  miles,  after  traversing  a 
distance  of  forty  or  fifty  miles,  with  an  average  velocity  of  about 
twenty-five  miles  per  second.     They  do  not  begin  to  be  visible  at  so 
great  an  elevation  as  the  aerolitic  meteors,  and  they  vanish  before 
they  penetrate  so  deeply  into  the  atmosphere. 

775.  Materials.  —  Occasionally  it  has  been  possible  to  catch  a 
glimpse  of  the  spectrum  of  one  of  the  brighter  shooting  stars,  and 
the  lines  of  sodium  and  magnesium  (probably)  are  quite  conspicuous, 
along  with  some  other  lines  which  cannot  be  securely  identified. 

As  these  bodies  are  completely  burned  up  before  they  reach  the 
earth,  all  we  can  ever  hope  to  get  of  their  material  is  the  product 
of  the  combustion.  In  most  places  the  collection  and  identifica- 
tion of  this  meteoric  ashes  is,  of  course,  hopeless  :  but  Norden- 

1  The  earth's  orbital  motion  is  always  directed  nearly  towards  the  point  on 
the  ecliptic  90°  west  of  the  sun. 

2  These  air  currents,  at  an  elevation  of  forty  miles  above  the  earth's  surface, 
are  thus  observed  to  have,  ordinarily,  velocities  of  from  50  to  75  miles  an  hour. 


474  METEORS. 

skiold  has  thought  he  might  find  it  in  polar  snows,  and  others  have 
thought  it  might  be  found  in  the  material  dredged  up  from  the 
bottom  of  the  ocean.  In  fact,  the  Swedish  naturalist,  by  melting 
several  tons  of  Spitzbergen  snow  and  filtering  the  water,  did  find  in 
it  a  sediment  containing  minute  globules  of  oxide  and  sulphide  of 
iron:  similar  globules  are  also  found  in  the  products  of  deep  sea 
dredging.  These  may  be  meteoric  ashes ;  but  it  is  quite  possible 
that  the  suspected  material  is  purely  terrestrial  in  its  origin. 

776.  Probable  Mass  of  Shooting  Stars.  —  We  have  no  very  certain 
means  of  getting  at  this.     We  can,  however,  fix  a  provisional  value 
by  means  of  the  amount  of  light  they  give.     Photometric  comparisons 
between  a  standard  star  and  a  meteor,  when  we  know  the  meteor's 
distance  and  the  duration  of  its  flight,  enable  us  to  ascertain  how 
the  total  amount  of  light  emitted  by  it  compares  with  that  given 
by  a  standard  candle  shining  for  one  minute.     Now,  according  to 
determinations    made    about    1860    by    Thomsen    at    Copenhagen 
(which  ought  to  be  repeated),  the  light  given  by  a  standard  candle  in 
a  minute  is  equivalent  to  about  twelve  foot-pounds  of  energy.     This 
excludes  all  the  energy  of  the  dark,  invisible  radiation  of  the  candle. 
Our  observations  of  the  meteor  give  us,  therefore,  its  total  luminous 
energy  in  foot-pounds ;   and  if  the  whole  of  the  meteor's  energy 
appeared  as  light,  then,  since  Energy  =  ^MV2,  we  could  at  once  get 
its  mass  by  dividing  twice  this  luminous  energy  by  the  square  of  the 
meteor's  velocity.      Since,  however,  only  a  small  portion  of  the 
meteor's  whole  energy  is  transformed  into  light,  the  mass  obtained 
in  this  way  would  be  too  small,  and  must  be  multiplied  by  a  factor 
which  expresses  the  ratio  between  the  total  energy  and  that  which 
is  purely  luminous.     It  is  not  likely  that  this  factor  exceeds  one 
hundred,  or  is  less  than  ten.     Assuming  the  largest  value,  however, 
the  photometric  observations  made  in  1866  and  1867  by  Professors 
Newcomb  and  Harkness  (stationed  respectively  at  Washington  and 
Richmond),  showed  that  the  majority  of  the  meteors  of  those  star- 
showers  weighed  less  than  a  single  grain.     The  largest  of  them  did 
not  reach  100  grains,  or  about  a  quarter  of  an  ounce.     A  similar 
result  follows  on  the  assumption  that  the  "luminous  efficiency" 
of  a  meteor  is  about  the  same  as  that  of  an  electric  incandescent 
lamp. 

777.  Growth  Of  the  Earth.  —  Since  the  earth  (in  fact,  every  planet) 
is  thus  continually  receiving  meteoric  matter,  and  sending  nothing  away 
from  it,  it  must  be  constantly  growing  larger;  but  this  growth  is  extremely 


EFFECT    ON   THE   EARTH'S    ORBIT.  475 

insignificant.  The  meteoric  matter  received  daily  by  the  earth,  if  we  accept 
one  grain  as  the  average  weight  of  a  shooting  star,  would  be  only  about 
a  ton,  after  making  a  reasonable  addition  for  occasional  aerolites.  If  we 
multiply  this  estimate  by  one  hundred,  it  certainly  will  be  exceedingly 
liberal,  and  at  that  rate  the  amount  received  by  the  earth  in  a  year  would 
amount  to  the  very  respectable  figure  of  36500  tons ;  and  yet,  even  at  this 
rate,  assuming  the  specific  gravity  of  the  average  meteor  as  three  times  that 
of  water,  it  would  take  about  1000  000000  years  to  accumulate  a  layer  one 
inch  thick  over  the  earth's  surface. 

778.  Effect  on  the  Earth's  Orbit.  —  Theoretically,  the  encounter  of 
the  earth  with  meteors  must  shorten  the  year  in  three  distinct  ways :  — 

First.  By  acting  as  a  resisting  medium,  and  so  diminishing  the  size 
of  the  earth's  orbit,  and  indirectly  accelerating  its  motion,  in  the  same 
manner  as  is  supposed  to  happen  with  Encke's  comet. 

Second.  By  increasing  the  attraction  between  the  earth  and  the  sun 
through  the  increase  of  their  masses. 

Third.  By  lengthening  the  day  —  the  earth's  rotation  being  made  slower 
by  the  increase  of  its  diameter,  so  that  the  year  will  contain  a  smaller  num- 
ber of  days. 

The  whole  effect,  however,  of  the  three  causes  combined,  does  not  amount 
to  ToVo  °f  a  second  in  a  million  of  years.  The  diminution  of  the  earth's 
distance  from  the  sun,  assuming  that  one  hundred  tons  of  meteoric  matter 
fall  daily,  and  also  assuming  that  the  meteors  are  moving  equally  in  all 
directions  with  the  parabolic  velocity  of  twenty-six  miles  per  second,  comes 
out  about  TJQ^QQ  of  an  inch  per  annum. 

Theoretically,  also,  the  same  meteoric  action  should  produce  a  shortening 
of  the  month,  and  Oppolzer  investigated  the  subject  a  few  years  ago,  to  see 
what  amount  of  meteoric  matter  would  account  for  the  observed  lunar  accel- 
eration (Art.  459).  He  found  that  it  would  require  an  amount  immensely 
greater  than  really  falls. 

779.  Meteoric  Heat:   Effect  of  Meteors  on  the  Transparency  of 
Space.  —  Of  course  each  meteor  brings  to  the  earth  a  certain  amount  of  heat 
developed  in  the  destruction  of  its  motion ;  and  at  one  time  it  was  thought 
that  a  very  considerable  percentage  of  the  total  heat  received  by  the  earth 
might  be  derived  from  this  source  (see  Art.  355  [2]).     Assuming,  how- 
ever, as  before,  the  fall  of  one  hundred  tons  of  meteoric  matter  daily  with 
an  average  velocity  of  twenty  miles  per  second  relative  to  the  earth,  the 
whole  amount  of  heat  comes  out  about  -^  calorie  per  annum  for  each  square 
metre  of  the  earth's  surface  —  as  much  in  a  year  as  the  sun  imparts  to  the 
same  surface  in  about  one-tenth  of  a  second. 

One  other  effect  of  meteoric  matter  in  space  should  be  alluded  to.  It 
must  necessarily  render  space  imperfectly  transparent,  like  a  thin  haze. 
Less  light  reaches  us  from  a  remote  star  than  if  the  meteors  were  absent. 


476 


METEORS. 


780.  Meteoric  Showers.  —  At  certain  times  the  shooting  stars, 
instead  of  appearing  here  and  there  in  the  sky  at  intervals  of  several 
minutes,  and  moving  in  all  directions,  appear  by  thousands,  and  even 
hundreds  of  thousands,  for  a  few  hours. 

The  Radiant.  —  At  such  times  they  do  not  move  without  system  ; 
but  they  all  appear  to  diverge  or  "  radiate  "  from  one  point  in  the 
sky;  that  is,  their  paths  produced  backward  all  intersect  at  a  common 
point  (or  nearly  so),  which  is  called  "the  radiant."  As  an  old  lady 
expressed  it,  in  speaking  of  the  meteoric  shower  of  1833,  "The  sky 
looked  like  a  great  umbrella."  The  meteors  which  appear  near  the 
radiant  are  stationary,  or  have  paths  extremely  short,  while  those 
which  appear  at  a  distance  from  it  have  long  courses.  The  radiant 


FIG.  210.  — The  Meteoric  Radiant  in  Leo,  Nov.  13, 1866. 

keeps  its  place  among  the  stars  unchanged,  during  the  whole  continu- 
ance of  the  shower,  and  the  shower  is  named  accordingly.  Thus  we 
have  the  meteor  shower  of  the  "Leonids"  whose  radiant  is  in  the 
constellation  of  Leo ;  similarly  the  "Andromedes  "  (or  Bielids),  the 
"Perseids,"  the  "  Geminids,"  the  "Lyrids,"  etc.  Fig.  210  is  a  chart 
of  the  tracks  of  meteors  observed  on  the  night  of  Nov.  13,  1866, 
showing  the  radiant  near  £  Leonis. 

The  simple  explanation  is  that  the  radiant  is  purely  an  effect  of 


DATES    OF    SHOWERS.  477 

perspective.  The  meteors  are  really  moving,  relatively  to  the  ob- 
server, in  lines  which  are  sensibly  straight  and  parallel,  as  are  also 
the  tracks  of  light  which  they  leave  in  the  air.  Hence  they  all  seem 
to  diverge  from  one  and  the  same  perspective  "vanishing  point." 
The  position  of  the  radiant  depends  entirely  upon  the  direction  of 
the  meteor's  motion  relative  to  the  earth. 

On  account  of  the  irregular  form  of  the  meteoric  particles,  they 
are  deflected  a  little  one  way  or  the  other  by  the  air  ;  neither  is  it 
likely  that  before  they  enter  the  air  their  paths  are  exactly  parallel. 
The  consequence  is  that  the  radiant,  instead  of  being  a  point,  is  an 
area  of  some  little  size,  usually  less  than  2°  in  diameter.  (For  note 
on  "  Stationary  Radiants/7  see  Art.  787*.) 

781.  Probably  the  most  remarkable  of  all  meteoric  showers  that  ever 
occurred  was  that  which  appeared  in  the  United  States  on  Nov.  12,  1833, 
in  the  early  morning  —  a  shower  of  Leonids.     The  number  that  fell  in 
the  five   or  six  hours   during  which  the   shower  lasted  was   estimated  at 
Boston  as  fully  250,000.     A  competent  observer  declared  that  "he  never 
saw  snow-flakes  thicker  in  a  storm  than  were  the  meteors  in  the  sky  at 
some   moments."     No  sound  was  heard,  nor  was  any  particle  known  to 
reach  the  earth. 

782.  Dates  of  Showers.  —  Since  the  meteor-swarm  pursues  a  regu- 
lar orbit  around  the  sun,  the  earth  can  only  encounter  it  when  she  is 
at  the  point  where  her  orbit  cuts  the  path  of  the  meteors  ;  and  this, 
of  course,  must  always  be  on  the  same  day  of  the  year,  except  as,  in 
the  process  of  time,  the  meteors'  orbits  slowly  shift  their  positions  on 
account  of  perturbations.     The  Leonid  showers,  therefore,  always 
appear  on  the  15th  of  November  (within  a  day  or  two)  ;  the  Androm- 
edes  on  the  27th  or  28th  of  the  same  month  ;  and  the  Perseids  early 
in  August. 

783.  Meteoric  Rings  and  Swarms. — If  the  meteors  are  scattered 
nearly  uniformly  around  their  whole  orbit,  so  as  to  form  a  ring,  the 
shower  will  recur  every  year;  but  if  the  flock  is  concentrated,  it  will  oc- 
cur only  when  the  meteor  group  is  at  the  meeting-place  at  the  same  time 
as  the  earth.    The  latter  is  the  case  with  the  Leonids  and  Andromecles. 
The  great  star-showers  from  these  groups  occur  only  rarely,  —  for  the 
Leonids  once  in  thirty-three  years,  and  for  the  Andromedes  (other- 
wise known  as  the  Bielids)  about  once  in  thirteen.     The  Perseids  are 
much  more  equally  and  widely  distributed,  so  that  they  appear  in 
considerable  numbers  every  year,  and  are  not  sharply  limited  to  a 


478  METEORS. 

particular  date,  but  are  more  or  less  abundant  for  a  fortnight  in  the 
latter  part  of  July  and  the  first  of  August, 

The  meteors  which  belong  to  the  same  group  all  have  a  resemblance  to 
each  other.  The  Perseids  are  yellowish,  and  move  with  medium  velocity. 
The  Leonids  are  very  swift,  for  we  meet  them  almost  directly,  and  they  are 
characterized  by  a  greenish  or  bluish  tint,  with  vivid  and  persistent  trains. 
The  Andromedes  are  sluggish  in  their  movements,  because  they  simply  over- 
take the  earth,  instead  of  meeting  it.  They  are  usually  decidedly  red  in 
color  and  have  only  small  trains.  About  100  "  meteoric  radiants  "  are  now 
recognized  and  catalogued.  The  most  conspicuous  (except  those  already 
named)  are  the  following  :  —  the  Draconids,  January  2  ;  Lyrids,  April  20  ; 
Aquariids  I,  May  6  ;  Aquariids  II,  July  28  ;  Orionids,  October  28  (see  Art. 
787*);  Geminids,  December  10. 

784.  The  Mazapil  Meteorite.  —  As  has  been  said,  during  these  showers 
no  sound  is  heard,  no  sensible  heat  perceived,  nor  do  any  masses  reach  the 
ground ;  with  the  one  exception,  however,  that  on  November  27,  1885,  a 
piece  of  meteoric  iron,  mentioned  in  the  list  given  in  Article  758,  fell  at 
Mazapil   in   Northern  Mexico  during   the   shower  of  Andromedes  which 
occurred  that  evening.     Whether  the  coincidence  is  accidental  or  not,  it  is 
interesting.     Many  high  authorities  speak  confidently  of  this  particular  iron 
meteor  as  being  really  a  piece  of  Biela's  comet  itself. 

785.  The  Connection  between  Comets  and  Meteors.  — At  the  time 
of   the   great   meteoric   shower   of    1833,   Professors   Olmsted   and 
Twining,  of  New  Haven,    recognized  the  fact  and  meaning  of   the 
radiant  as  pointing  to  the  existence  of  swarms  of  meteoric  particles 
revolving  in  regular  orbits  around  the  sun  ;  and  Olmsted  at  the  time 
went  so  far  as  even  to  call  the  body  or  swarm  a  "  comet.*'     In  some 
respects,  however,  his  views  were  seriously  wrong,  and  soon  received 
modification  and  correction  from  other  astronomers.      Erman  espe- 
cially pointed  out  that  in  some  cases,  at  least,  it  would  be  necessary 
to  suppose  that  the  meteors  were  distributed  in  rings,  and  he  also 
developed  methods  by  which  the  meteoric  orbits  could  be  computed 
if   the   necessary  data   could   be   secured.      Olmsted   and  Twining, 
however,  were  the  first  to  show  that  the  meteors  are  not  terrestrial 
and  atmospheric,  but  bodies  truly  cosmical. 

The  subject  was  taken  up  later  by  Professor  Newton,  of  New 
Haven,  who  in  1864  showed  by  an  examination  of  old  records  that 
there  had  been  a  number  of  great  autumnal  meteoric  star-showers  at 
intervals  of  just  about  thirty-three  years,  and  he  predicted  confidently 
a  shower  for  Nov.  13-14,  1866.  As  to  the  orbit  of  the  meteoric 
body  (or  ring,  according  to  Erman's  view) ,  he  found  that  it  might, 


CONNECTION   BETWEEN    COMETS   AND   METEOBS. 


479 


consistently  with  what  had  been  so  far  observed,  have  either  of  Jive 
different  orbits  ;  one  with  a  period  of  33 £  years,  two  with  periods  of 
one  year  ±11  days,  and  two  with  periods  of  half  a  year  ±  51  days. 
He  considered  rather  most  probable  the  period  of  354  days;. but  he 
pointed  out  that  the  slow  change  that  had  taken  place  in  the  annual 
date  of  the  shower l  would  furnish  the  means  of  determining  which 
of  the  orbits  was  the  true  one. 

This  change  of  date  indicates  a  slow  motion  of  the  nodes  of  the 
orbit  of  the  meteoric  body  at  the  rate  of  about  52"  a  year.  Adams, 
of  Neptunian  fame,  made  the  laborious  calculation  of  the  effect  of 
planetary  perturbations  upon  each  of  the  five  different  orbits  sug- 
gested by  Professor  Newton,  and  showed  that  the  true  orbit  must  be 
the  largest  one  which  has  a  period  of  331  years. 


FIG.  211.  —  Orbits  of  Meteoric  Swarms  which  are  known  to  be  associated  with  Comets. 

The  meteoric  shower  occurred  in  1866  as  predicted,2  and  was 
repeated  in  1867,  the  meteor-swarm  being  stretched  out  along  its 
orbit  for  such  a  distance  that  the  procession  is  nearly  three  years 
in  passing  any  given  point. 

1  In  A.D.  902  (the  "year  of  the  stars"  in  the  old  Arab  chronicles),  the  date 
was  what  would  be  Oct.  19,  in  our  "  new  style  "  reckoning.    In  1202  the  shower 
occurred  five  days  later,  and  in  1833  the  date  was  Nov.  12. 

2  See  note  on  page  482. 


480 


METEOKS. 


786.  Identification  of  Cometary  and  Meteoric  Orbits. — The  re- 
searches of  Newton  and  Adams  had  awakened  lively  interest  in 
the  subject,  and  Schiaparelli,  of  Milan,  a  few  weeks  after  the  Leonid 
shower,  published  a  paper  upon  the  Perseids,  or  August  meteors,  in 
which  he  brought  out  the  remarkable  fact  that  they  were  moving  in 
the  same  path  as  that  of  the  bright  comet  of  1862,  known  as  Tuttle's 
Comet.  Shortly  after  this  Leverrier  published  his  orbit  of  the 
Leonid  meteors,  derived  from  the  observed  position  of  the  radiant  in 
connection  with  the  periodic  time  assigned  by  Adams ;  and  almost 
simultaneously,  but  without  any  idea  of  a  connection  between  them, 
Oppolzer  published  his  orbit  of  TempePs  comet  of  1866  ;  and  the 
two  orbits  were  at  once  seen  to  be  practically  identical.  Now  a  single 
case  of  such  a  coincidence  as  that  pointed  out  by  Schiaparelli,  might 


PIG.  212.  —  Transformation  of  the  Orbit  of  the  Leonids  by  the  Encounter  with  Uranus,  A.D.  126. 


possibly  be  accidental,  but  hardly  two.  Then  five  years  later,  in 
1872,  came  the  meteoric  shower  of  the  Andromedes,  following  in  the 
track  of  Biela's  comet ;  and  among  the  more  than  one  hundred  dis- 
tinct meteor-swarms  now  recognized,  Professor  Alexander  Herschel 
finds  four  or  five  others  which  have  a  "  comet  annexed,"  so  to  speak. 
Fig.  211  represents  the  orbits  of  four  of  the  meteoric  swarms  which 
are  known  to  be  associated  with  comets. 


IDENTIFICATION  OF  COMETARY  AND  METEORIC  ORBITS.      481 

787.  In  the  cases  of  the  Leonids  and  Andromedes  the  meteor-swarm 
follows  the  comet.  Many  believe,  however,  that  the  comet  itself  is  simply 
the  thickest  part  of  the  swarm.  Kirkwood  and  Schiaparelli  have  both 
pointed  out  that  a  body  constituted  as  a  comet  is  supposed  to  be,  must 
almost  necessarily. break  up  in  consequence  of  the  "  tide-producing  "  pertur- 
bations of  the  sun,  independent  of  any  repulsive  action  such  as  is  supposed 
to  be  the  cause  of  a  comet's  tail.  They  hold  that  these  meteor-swarms  are 
therefore  merely  the  product  of  a  comet's  disintegration. 

The  longer  the  comet  has  been  in  the  system,  the  more  widely  scattered 
will  be  its  particles.  The  Perseids  are  supposed,  therefore,  to  be  old  inhab- 
itants of  the  solar  system,  while  the  Leonids  and  Andromedes  are  compara- 
tively new-comers.  Leverrier  has  shown  that  in  the  year  A.D.  126  Tempers 
comet  must  have  been  very  near  to  Uranus,  and  a  natural  inference  is  that 
it  was  introduced  into  the  solar  system  at  that  time.  Fig.  212  illustrates  his 
hypothesis.  However  these  things  may  be,  it  is  now  certain  that  the  connec- 
tion between  comets  and  meteors  is  a  very  close  one,  though  it  can  hardly 
be  considered  certain  as  yet  that  every  scattered  group  of  meteors  is  the 
result  of  cometary  disintegration.  We  are  not  sure  that  when  a  cometary 
mass  first  enters  the  solar  system  from  outer  space,  it  comes  in  as  a  close- 
packed  swarm. 

787*.  Stationary  Radiants.  —  When  a  meteoric  shower  persists  for 
days  and  even  weeks,  as  is  the  case  with  the  Perseids,  for  instance,  the  radiant 
as  a  rule  gradually  shifts  its  position  among  the  stars  on  account  of  the 
change  in  the  direction  of  the  earth's  motion  during  the  time  —  as  it  ought 
to,  since  the  place  of  the  radiant  depends  upon  the  combination  of  the 
earth's  motion  with  that  of  the  meteors. 

But  Mr.  Denning,  of  Bristol  (England),  for  many  years  an  assiduous 
observer  of  meteors,  claims  to  have  discovered  numerous  cases  in  which  the 
radiant  of  a  long-continued  shower  remains  stationary;  and  he  presents  as 
typical  the  Orionids,  which  scatter  along  from  about  October  10  to  the  24th, 
all  the  time,  according  to  his  observations,  keeping  their  radiant  close  to  the 
star  v  Orionis.  Only  doubtful  explanations  of  such  fixity  have  yet  appeared, 
and  though  Mr.  Denning  is  perfectly  confident  of  the  genuineness  of  his 
discovery,  and  though  it  is  very  generally  accepted  as  a  fact,  some  very  high 
authorities,  Tisserand,  for  instance,  have  questioned  it,  as  being  "  incredible 
and  unaccountable." 


EXERCISES  ON  CHAPTER  XIX. 

1.  If  a  compact  swarm  of  meteors  were  now  to  enter  the  system  and  be 
deflected  by  the  attraction  of  some  planet  into  an  elliptical  orbit  around 
the  sun,  would  the  swarm  continue  to  be  compact?  If  not,  what  would  be 
the  ultimate  distribution  of  the  meteors? 


4:82  EXERCISES. 

2.  What  is  the  probable  relative  age  of  meteoric  swarms  and  meteoric 
rings  as  members  of  the  solar  system? 

3.  Assuming  that  the  earth  encounters  twenty  million  meteors  every 
twenty-four  hours,  what  is  the  average  number  in  a  cubic  space  of  a 
thousand  million  cubic  miles,  (i.e.,  a  cube  a  thousand  miles  on  each  edge)  ? 

Ans.   About  250. 

4.  .  If  space  were  occupied  by  meteors  uniformly  distributed  a  hundred 
miles  apart  on  three  sets  of  lines  perpendicular  to  each  other,  how  many 
would  be  encountered  by  the  earth  in  a  day? 

Ans.    78700000. 

NOTE.  — In  this  cubical  arrangement  the  average  distance  between  the  meteors  much 
exceeds  100  miles.  If  they  were  packed  as  closely  as  possible,  consistently  with  the  con- 
dition that  the  distance  between  two  neighbors  should  nowhere  be  less  than  100  miles,  the 
number  would  be  increased  by  nearly  forty  per  cent. 

NOTE  TO  ART.  785. 

The  shower  of  Leonids,  which  was  expected  in  November,  1899,  failed  to 
appear.  The  preceding  year  Leonids  were  observed  in  considerable  numbers, 
indicating  that  the  meteoric  swarni  was  nearing  us,  but  in  1899  very  few  were 
seen,  though  carefully  watched  for.  The  cause  of  the  failure  is  not  yet  quite 
certain;  it  was,  however,  probably  due  to  considerable  perturbations  of  the 
meteoric  orbit  during  the  past  thirty-three  years,  caused  by  the  action  of  the  outer 
planets,  especially  Saturn.  It  seems  not  unlikely  that  the  effect  has  been  to 
shift  the  plane  of  this  orbit  so  as  to  abolish  its  former  "grade-crossing"  with 
the  orbit  of  the  earth,  causing  the  meteors  to  pass  above  or  below  our  level. 

But  the  mathematical  difficulties  of  the  problem  are  enormous,  owing  to  the 
great  extent  of  the  meteoric  flock,  and  the  results  of  calculation  are  therefore 
somewhat  uncertain.  It  is  quite  possible  that  we  may  yet  run  into  the  Leonids 
a  year  or  two  later. 

1904,  —  The  possibility  indicated  above  was  realized  on  the  mornings  of  No- 
vember 14  to  15,  both  in  1901  and  1902,  and  also  to  some  extent  in  1903.  The 
Leonids  appeared  in  considerable  numbers,  though  no  one  of  the  displays  was  at 
all  comparable  with  the  showers  of  1866-7,  not  to  speak  of  1833.  In  1901  the 
meteors  were  visible  for  the  most  part  only  west  of  the  Mississippi ;  in  1902  and 
1903,  in  Europe.  TeinpePs  comet,  if  it  returned  in  1900,  escaped  observation. 


THE    STARS  :     THEIR    NATURE   AND    NUMBER.  483 


CHAPTER    XX. 

THE  STARS :  THEIR  NATURE  AND  NUMBER.  —  THE  CONSTEL- 
LATIONS. —  STAR-CATALOGUES.  —  DESIGNATION  AND  NOMEN- 
CLATURE. —  PROPER  MOTIONS  AND  THE  MOTION  OF  THE  SUN 
IN  SPACE.  —  STELLAR  PARALLAX  AND  DISTANCE. 

788.  WE  enter  now  upon  a  vaster  subject.     Leaving  the  con- 
fines of  the  solar  system  we  cross  the  void  that  makes  an  island1 
of  the  sun's  domains,  and  enter  the  universe  of  the  stars.     The 
nearest  star,  so  far  as  we  have  yet  been  able  to  ascertain,  is  one 
whose  distance  is  more  than  250000  times  the  radius  of  the  earth's 
annual  orbit ;  so  remote  that,  seen  from  that  star,  the  sun  itself 
would  appear  only  about  as  bright  as  the  pole  star,  and  from  it  no 
telescope  ever  yet  constructed  could  render  visible  a  single  one 
of  all  the  retinue  of  planets  and  comets  that  make  up  the  solar 
system. 

789.  Nature  of  the  Stars.  —  As  shown  by  their  spectra  the  stars 
are  suns ;  that  is,  they  are  bodies  comparable  in  magnitude  and  in 
physical  condition  with  our  own  sun,  shining  by  their  own  light  as 
the  sun  does,  and  emitting  a  radiance  which  in  many  cases  could 
not   be  distinguished   from  sunlight  by  any  of   its  spectroscopic 
characteristics.     Some  of  them  are  vastly  larger  and  hotter  than 
our  sun,  others  smaller  and  cooler,  for,  as  we  shall  see,  they  differ 
enormously  among  themselves. 

790.  Number  of  the  Stars.  —  The  impression  on  a  dark  night  is  of 
absolute  countlessness  ;  but,  in  fact,  the  number  visible  to  the  naked 
eye  is  very  limited,  as  one  can  easily  discover  by  taking  some  definite 
area  in  the  sky,  say  the  "  bowl  of  the  dipper,"  and  counting  the  stars 
which  he  can  see  within  it.     He  will  find  that  the  number  which  he 

1  That  the  solar  system  is  thus  isolated  by  a  surrounding  void  is  proved  by 
the  almost  undisturbed  movements  of  Uranus  and  Neptune ;  for  their  pertur- 
bations would  betray  the  presence  of  any  body,  at  all  comparable  with  the  sun 
in  magnitude,  within  a  distance  a  thousand  times  as  great  as  that  between  the 
earth  and  sun. 


484  THE   STABS, 

can  fairly  count  is  surprisingly  small,  though  by  averted  vision  he  will 
get  uncertain  glimpses  of-  many  more.  In  the  whole  celestial  sphere 
the  number  bright  enough  to  be  visible  to  the  naked  eye  is  only  from 
6000  to  7000  in  a  clear,  moonless  sky.  A  little  haze  or  moonlight 
cuts  out  fully  half  of  them,  and  of  course  there  is  a  great  difference 
in  eyes.  But  the  sharpest  eyes  could  probably  never  fairly  see  more 
than  2000  or  3000  at  one  time,  since  near  the  horizon  the  smaller  stars 
are  invisible,  and  they  are  immensely  the  most  numerous,  fully  half 
of  the  whole  number  being  those  which  are  just  on  the  verge  of  visi- 
bility. The  total  number  that  can  be  seen  well  enough  for  observation 
with  such  instruments  as  were  used  before  the  invention  of  the  telescope 
is  not  quite  1100. 

With  even  a  small  telescope  the  number  is  enormously  increased. 
A  mere  opera-glass  an  inch  and  a  half  in  diameter  brings  out  at  least 
100,000.  The  telescope  with  which  Argelander  made  his  Durchmus- 
terung  of  more  than  300,000  stars  —  all  north  of  the  celestial  equator 
—  had  a  diameter  of  only  two  inches  and  a  half.  The  number  visible 
in  the  great  Lick  l  telescope  of  three  feet  diameter  is  probably  nearly 
100,000000. 

791.  Constellations. — In  ancient  times  the  stars  were  grouped  by 
"constellations,"  or  "  asterisms,"  partly  as  a  matter  of  convenient 
reference  and  partly  as  superstition.  Man}  of  the  constellations  now 
recognized,  — all  of  those  in  the  zodiac  and  those  about  the  northern 
pole, — >are  of  prehistoric  antiquity.  To  these  groups  were  given  fanci- 
ful names,  mostly  of  persons  or  objects  conspicuous  in  the  mythological 
records  of  antiquity  ;  a  great  number  of  them  are  connected  in  some 
way  or  other  with  the  Argonautic  expedition. 

In  some  cases  the  eye  can  trace  in  the  arrangement  of  the  stars  a  vague 
resemblance  to  the  object  which  gives  name  to  the  constellation ;  but  gener- 
ally no  reason  can  be  assigned  why  the  constellation  should  be  so  named  or 
so  bounded.  Of  the  sixty-seven  constellations  now  usually  recognized  on 
celestial  globes,  forty-eight  have  come  down  from  Ptolemy.  The  others 
have  been  formed  by  Hevelius,  Bayer,  Royer,  and  one  or  two  other  astron- 
omers, to  embrace  stars  not  included  in  Ptolemy's  constellations,  and  espe- 
cially to  furnish  a  nomenclature  for  the  stars  never  seen  by  Ptolemy  on 
account  of  their  nearness  to  the  southern  pole.  A  considerable  number  of 

1  Neglecting  the  loss  of  light  in  the  lenses,  the  Lick  telescope  ought,  theoreti- 
cally, to  show  stars  so  faint  that  it  would  take  more  than  30,000  of  them  to  make 
a  star  equal  to  the  faintest  that  can  be  seen  with  the  naked  eye.  (See  Art.  38.) 


LIST   OF   CONSTELLATIONS.  485 

other  constellations,  which  have  been  tentatively  established  at  various 
times,  and  are  sometimes  found  on  globes  and  star-maps,  have  been  given 
up  as  useless  and  impertinent. 

792.  We  present  a  list  of  the  constellations,  omitting,  however,  some 
of  the  modern  ones  which  are  now  not  usually  recognized  by  astronomers. 
The  constellations  are  arranged  both  vertically  and  horizontally.  The  order 
in  the  vertical  columns  is  determined  by  right  ascension,  as  indicated  by 
the  Roman  numbers  at  the  left.  Horizontally  the  arrangement  is  according 
to  distance  from  the  north  pole,  as  shown  by  the  headings  of  the  columns. 
The  number  appended  to  each  constellation  gives  the  number  of  stars  it 
contains,  down  to  and  including  the  6th  magnitude.  The  zodiacal  constel- 
lations are  italicized,  and  the  modern  constellations  are  marked  by  an 
asterisk. 

The  different  groups  of  constellations  are  found  near  the  meridian  at 
half-past  eight  o'clock,  P.M.,  on  the  dates  indicated  below. 

Group  (I.,  II.),  Dec.  1.     These  constellations  contain  no  first-magnitude 

stars,  but  Cassiopeia,  Andromeda,  Aries,  and  Cetus  include  enough 

stars  of  the  second  and  third  magnitude  to  be  fairly  conspicuous. 
Group  (III.,  IV.),  Jan.  1.     Perseus  north  of  the  zenith,  and  the  Pleiades 

and  Aldebaran  in  Taurus,  are  characteristic. 
Group  (V.,  VI.),  Feb.  1.     On  the  whole  this  is  the  most  brilliant  region 

of  the  sky  and  Orion  the  finest  constellation. 
Group  (VII.,  VIII.),  March  1.     Characterized  by  Procyon  and  Sirius,  the 

latter  incomparably  the  brightest  of  all  the  fixed  stars. 
Group  (IX.,  X.),  April  1.     Leo  is  the  only  conspicuous  constellation. 
Group  (XI.,  XII.),  May  1.     A  barren  region,  except  for  Ursa  Major  north 

of  the  zenith. 
Group  (XIII.,  XIV.),  June  1.     Marked  by  Arcturus,  the  brightest  of  the 

northern  stars,  with  the  paler  Spica  south  of  the  equator. 
Group  (XV.,  XVI.),  July  1.     The  Northern  Crown  and  Hercules  are  the 

most  characteristic  configurations. 
Group  (XVII.,  XVIII.),  Aug.  1.     Vega  is  nearly  overhead,  and  the  red 

Antares  low  down  in  the  south,  with  Altair  near  the  equator,  just 

east  of  Ophiuchus. 
Group  (XIX.,  XX.),  Sept.  1.     Cygnus  is  in  the  zenith,  and  Sagittarius 

low  down,  while  the  brightest  part  of  the  Milky  Way  lies  athwart 

the  meridian. 
Group  (XXI.,  XXII.),  Oct.  1.     A  barren  region,  relieved  only  by  the 

bright  star  Fomalhaut  of  the  Southern  Fish  near  the  southern 

horizon. 
Group  (XXIII.,  XXIV.),  Nov.  1.     This  region    also   is   rather  barren, 

though  the  "  great  square  "  of  Pegasus  is  a  notable  configuration 

of  stars. 


486 


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.      DESIGNATION   OF   BRIGHT   STARS.  487 

793,  A  thorough  knowledge  of  these  artificial  groups,  and  of  the 
names  and  locations  of  the  stars  in  them,  is  not  at  all  essential,  even 
to  an  accomplished  astronomer  ;  but  it  is  a  matter  of  very  great  con- 
venience to  know  the  principal  constellations,  and  perhaps  a  hundred 
of  the  brightest  stars,  well  enough  to  be  able  to  recognize  them 
readily  and  to  use  them  as  points  of  reference.     This  amount  of 
knowledge   is   easily  acquired  by  three  or  four  evenings'  study  of 
the  sky  in  connection  with  a  good  star-map  or  celestial  globe,  taking 
care  to  observe  on  evenings  at  different  seasons  of  the  year,  so  as  to 
command  the  whole  sky. 

At  present  the  best  star-afclas  for  reference  is  probably  that  of  Klein. 
The  maps  of  Argelander's  "  Uranometria  Nova  "  and  Heis's  atlas  (both  in 
German)  are  handsomer,  and  for  some  purposes  more  convenient.  There 
are  many  others,  also,  which  are  excellent.  The  smaller  maps  which  are 
found  in  the  text-books  on  astronomy  are  not  on  a  scale  sufficiently  large  to 
be  of  much  scientific  use,  though  they  answer  well  enough  the  purpose  of 
introducing  the  student  to  the  principal  star-groups.  A  new  and  excellent 
atlas  by  Upton  has  recently  been  published  by  Ginn  &  Co. 

794.  Designation  of  Bright  Stars.  —  (a)  Names.     Some  fifty  or 
sixty  of  the  brighter  stars  have  names  of  their  own  in  common  use- 
A  majority  of  the  names  belonging  to  stars  of  the  first  magnitude  are 
of  Greek  or  Latin  origin,  and  significant,  as,  for  instance,  Arcturus, 
Sirius,  Procyon,  Regulus,  etc.     Some  of  the  brightest  stars,  however, 
have  Arabic  names,  as  Aldebaran,  Vega,  and  Betelgeuse,  and  the 
names  of  most  of  the  smaller  stars  are  Arabic,  when  they  have  names 
at  all. 

(b)  Place  in  Constellation.      Spica  is  the  star  in  the  handful  of 
wheat  carried  by  Virgo ;  C}Tnosure  signifies  the  star  at  the  end  of  the 
Dog's  Tail    (in   ancient  times  the  constellation  we  now  call  Ursa 
Minor  seems    to   have  been    a   dog)  ;    Capella  is   the   goat   which 
Auriga,  the  charioteer,  carries  in  his  arms.     Hipparchus,  Ptolemy, 
and,  in  fact,  all  the  older  astronomers,  including  Tycho  Brahe,  used 
this  clumsy  method  almost  entirely  in  designating  particular  stars; 
speaking,  for  instance,  of  the  star  in  the  "  head  of  Hercules,"  or  in 
the  "  right  knee  of  Bootes,"  and  so  on. 

(c)  Constellation  and  Letters.     In  1603  Bayer,  in  publishing  a  new 
star-map,  adopted  the  excellent  plan,  ever  since  in  vogue,  of  designat- 
ing the  stars  in  the  different  constellations  by  the  letters  of  the  Greek 
alphabet,  assigned  usually  in  order  of  brightness.     Thus  Aldebaran  is 


488  THE    STABS. 

a  Tauri,  the  next  brightest  star  in  the  constellation  is  fi  Tauri,  and 
so  on,  as  long  as  the  Greek  letters  hold  out ;  then  the  Roman  letters 
are  used  as  long  as  they  last ;  and  finally,  whenever  it  is  found  neces- 
sary, we  use  the  numbers  which  Flamsteed  assigned  a  century  later. 
At  present  every  naked-eye  star  can  be  referred  to  and  identified  by 
some  letter  or  number  in  the  constellation  to  which  it  belongs. 

(d)  Current  Number  in  a  Star- Catalogue.  Of  course  all  the  above 
methods  fail  for  the  hundreds  of  thousands  of  smaller  stars.  In  their 
case  it  is  usual  to  refer  to  them  as  number  so-and-so  of  some  well- 
known  star-catalogue;  as,  for  instance,  22,500  LI.  (Lalande),  or 
2573  B.  A.  C.  (British  Association  Catalogue) .  At  present  our  various 
star-catalogues  contain  from  600,000  to  800,000  stars,  so  that,  except 
in  the  Milky  Way,  almost  any  star  visible  in  a  telescope  of  two  or 
three  inches'  aperture  can  be  identified  and  referred  to  by  means  of 
some  star-catalogue  or  other. 

Synonyms.  Of  course  all  the  brighter  stars  which  have  names  have 
also  letters,  and  are  sure  to  be  included  in  every  star- catalogue  which 
covers  their  part  of  the  sky.  A  given  star,  therefore,  has  often  a 
large  number  of  aliases,  and  in  dealing  with  the  smaller  stars  great 
pains  must  be  taken  to  avoid  mistakes  arising  from  this  cause. 

STAR-CATALOGUES. 

795.  These  are  lists  of  stars  arranged  in  regular  order  (at  present 
usually  in  order  of  right  ascension) ,  and  giving  the  places  of  the  stars 
at  some  given  epoch,  either  by  means  of  their  right  ascensions  and 
declinations,  or  by  their  (celestial)  latitudes  and  longitudes.  The 
so-called  "magnitude,"  or  brightness  of  the  star,  is  also  ordinarily 
indicated.  The  first  of  these  star-catalogues  was  that  of  Hipparchus, 
containing  1080  stars  (all  that  are  easily  visible  and  measurable  by 
naked-eye  instruments) ,  and  giving  their  longitudes  and  latitudes  for 
the  epoch  of  125  B.C. 

This  catalogue  has  been  preserved  for  us  by  Ptolemy  in  the  Almagest, 
and  from  it  he  formed  his  own  catalogue,  reducing  the  positions  of  the 
stars  (i.e.,  correcting  for  precession  the  positions  given  by  Hipparchus)  to 
his  own  epoch,  about  150  A.D.  •  The  next  of  the  old  catalogues  of  any  value 
is  that  of  Ulugh  Beigh  made  at  Samarcand  about  1450  A.D.  This  appears 
to  have  been  formed  from  independent  observations.  It  was  followed  in 
1580  by  the  catalogue  of  Tycho  Brahe  containing  1005  stars,  the  last  which 
was  constructed  before  the  invention  of  the  telescope. 

The  modern  catalogues  are  numerous.  Some  give  the  places  of  a  great 
number  of  stars  rather  roughly,  merely  as  a  means  of  identifying  them  when 


STAR-CATALOGUES.  489 

used  for  cometary  observations  or  other  similar  purposes.  To  this  class 
belongs  Argelander's  Durchmusterung  of  the  northern  heavens,  which  con- 
tains over  324000  stars,  —  the  largest  number  in  any  one  catalogue  thus  far 
published.  This  has  since  been  supplemented  by  Schoenfeld's  southern 
Durchmusterung  on  a  similar  plan.  Then  there  are  the  "  catalogues  of  pre- 
cision,'" like  the  Pulkowa  and  Greenwich  catalogues,  which  give  the  places 
of  a  few  hundred  stars  as  accurately  as  possible  in  order  to  furnish  "  funda- 
mental stars,"  or  reference  points  in  the  sky.  The  so-called  "  Zones "  of 
Bessel,  Argelander,  Gould  and  many  others,  are  catalogues  covering  limited 
portions  of  the  heavens,  containing  stars  arranged  in  zones  about  a  degree 
wide  in  declination,  and  running  some  hours  in  right  ascension.  An  immense 
cooperative  catalogue  is  now  in  process  of  publication  under  the  auspices  of 
the  German  Astronomische  Gesellschaft,  and  will  contain  accurate  places  of 
all  stars  above  the  9th  magnitude  north  of  15°  south  declination.  The  ob- 
servations are  practically  completed,  and  most  of  the  "  parts "  have  already 
appeared.  (See  also  Art.  798.) 

t 

796.  Determination  of  Star-Places.  —  The  observations  from  which 
a  star-catalogue  is  constructed  are  usually  made  with  the  meridian 
circle  (Art.  63).    For  the  catalogues  of  precision,  comparatively  few 
stars  are  observed,  but  all  with  the  utmost  care  and  during  several 
years,  taking  all  possible  means  to  eliminate  instrumental  and  obser- 
vational errors  of  every  sort. 

In  the  more  extensive  catalogues  most  of  the  stars  are  observed  only 
once  or  twice,  and  everything  is  made  to  depend  upon  the  accuracy 
of  the  places  of  the  fundamental  stars,  which  are  assumed  as  correct. 
The  instrument  in  this  case  is  used  only  "  differentially  "  to  measure 
the  comparatively  small  differences  between  the  right  ascension  and 
declination  of  the  fundamental  stars  and  those  of  the  stars  to  be  cata- 
logued. 

797.  Method  of  using  a  Catalogue.  — The  catalogue  contains  the 
mean  right  ascension  and  declination  of  its  stars  for  the  beginning 
of  some  given  year ;  i.e.,  the  right  ascension  and  declination  the  star 
would  have  at  that  time  if  there  were  no  aberration  of  light  and  no 
irregular  motion  of  the  celestial  pole  to  affect  the  position  of  the 
equator  and  equinox.     To  determine  the  actual  apparent  right  ascen- 
sion and  declination  of  a  star  for  a  given  date  (which  is  what  we 
want  in  practice),  the  catalogue  place  must  be  "reduced"  to  the 
date  in  question  ;  i.e.,  it  must  be  corrected  for  precession,  nutation, 
and  aberration. 

The  operation  with  modern  tables  and  formulas  is  not  a  very  tedious  one, 
involving  perhaps  five  minutes'  work,  but  without  it  the  catalogue  places  are 


490 


THE    STARS. 


FIG.  213.— The  Photographic  Telescope  of  the  Henry  Brothers,  Paris. 


STAR-CHARTS.  491 

useless  for  most  purposes.  Vice  versa,  the  observations  of  a  fixed  star  with 
the  meridian  circle  do  not  give  its  mean  right  ascension  and  declination 
ready  to  go  into  the  catalogue,  but  the  observations  must  be  reduced  from 
apparent  place  to  mean  before  they  can  be  tabulated. 

798.  Star-Charts.  —  For  many  purposes  charts  of  the  stars  are 
more  convenient  than  a  catalogue,  as,  for  instance,  in  searching  for 
new  planets.  The  old-fashioned  way  of  making  such  charts  was  by 
plotting  the  results  of  zone  observations.  The  modern  way,  intro- 
duced within  the  last  few  years,  is  to  do  it  by  photography,  the 
cardinal  advantages  being  two :  first,  that  in  this  way  a  great 
number  of  stars  can  be  automatically  and  permanently  registered 
at  one  operation,  and  afterwards  studied  and  measured  at  leisure ; 
second,  that  by  a  sufficient  prolongation  of  the  exposure  stars  far 
too  faint  to  be  seen  by  the  telescope  used  can  be  made  to  impress 
themselves  upon  the  plate.  The  plan  decided  upon  at  the  Paris 
Astronomical  Congress  in  1887  contemplates  the  photographing 
of  the  whole  sky  upon  glass  plates  about  eight  inches  square,  each 
covering  an  area  of  2°  square  (four  square  degrees),  showing  all  stars 
down  to  the  fourteenth  magnitude.  The  enterprise  is  now  (1908)  well 
advanced,  fully  three-quarters  of  the  negatives  having  been  already 
made.  Fourteen  different  observatories  have  cooperated  in  the 
work,  only  one  of  them,  however,  in  America, — the  Mexican  National 
Observatory  at  Chepultepec. 

The  figure  (Fig.  213)  is  a  representation  of  the  Paris  instrument  of  the 
Henry  Brothers,  which  was  adopted  as  the  typical  instrument  for  the  opera- 
tion. It  has  an  aperture  of  about  fourteen  inches,  and  a  length  of  about 
eleven  feet,  the  object-glass  being  specially  corrected  for  the  photographic 
rays.  A  9-inch  visual  telescope  is  enclosed  in  the  same  tube  so  that  the 
observer  can  watch  the  position  of  the  instrument  during  the  whole 
operation. 

The  other  instruments  differ  in  mechanical  arrangements,  but  all  have 
lenses  of  the  same  aperture  and  focal  length,  the  scale  of  all  the  negatives 
being  1'  to  a  millimetre,  —  the  same  as  that  of  Argelander's  charts. 

It  was  originally  planned  to  give  each  plate  20  minutes'  exposure,  but 
improvements  in  the  photographic  plates  since  the  meeting  of  the  Congress 
now  make  it  possible  to  cut  down  the  time  very  materially.  It  will  require 
about  11000  plates  of  the  size  named  to  cover  the  whole  sky,  and  as  each 
star  is  to  appear  on  two  plates  at  least,  the  whole  number  of  plates,  allowing 
for  overlaps,  will  be  about  22000.  As  every  plate  will  contain  upon  it  a 
number  of  well-determined  catalogue  stars,  it  will  furnish  the  means  of 
determining  accurately,  whenever  needed,  the  place  of  any  other  star  which 
appears  upon  the  same  plate. 


492  THE   STAKS. 

The  places  of  all  stars  above  the  twelfth  magnitude  are  to  be  carefully 
measured  upon  the  plates,  and  will  furnish  an  enormous  catalogue,  containing 
at  least  2  000000  stars. 

798*.  Several  other  very  large  photographic  telescopes  have 
recently  been  constructed.  The  Bruce  telescope  (presented  to  the 
Harvard  College  Observatory  by  Miss  Bruce  of  New  York)  has  a 
four-lens  objective  two  feet  in  diameter,  but  with  a  focal  length  of 
only  11  feet,  the  same  as  those  mentioned  above ;  so  that  its  nega- 
tives will  be  on  the  same  scale. 

It  has  been  sent  to  Arequipa  (Peru),  where  it  is  now  employed  in 
the  photography  (and  spectroscopy)  of  the  southern  heavens.  The 
new  photographic  telescope  at  Greenwich  has  the  same  aperture, 
but  is  much  longer ;  and  an  instrument  similar  to  it  has  been  re- 
cently mounted  at  the  Cape  of  Good  Hope.  Both  have  visual  "  find- 
ers "  18  inches  in  diameter.  The  enormous  instrument  at  Meudon 
(near  Paris)  has  also  two  telescopes  combined, —  a  visual  telescope 
of  32  inches  aperture,  and  a  photographic  of  25  inches,  each  55  feet 
focal  length.  But  these  long-focus  instruments  will  be  used  mainly 
for  other  purposes  than  charting.  A  huge  instrument  of  31|-  inches 
aperture  has  still  more  recently  (1899)  been  mounted  at  Potsdam. 

STAR    MOTIONS. 

799.  The  stars  are  ordinarily  called  "fixed,"  in  distinction  from 
the  planets  or  "  wanderers,"  because  as  compared  with  the  sun  and 
moon  and  planets  they  have  no  evident  motion,  but  keep  their  relative 
positions  and  configurations  unchanged.    Observations  made  at  suffi- 
ciently wide  intervals  of  time,  and  observations  with  the  spectroscope, 
show,  however,  that  they  are  really  moving,  and  that  with  velocities 
which  are  comparable  to  the  motion  of  the  earth  in  her  orbit. 

If  we  compare  the  right  ascension  and  declination  of  a  star  deter- 
mined to-day  with  that  determined  a  hundred  years  ago,  they  will 
be  found  different.  The  difference  is  mainly  due  to  precession  and 
nutation,  which  are  not  motions  of  the  stars  at  all,  but  simply 
changes  in  the  position  of  the  reference  circles  used,  and  due  to 
alterations  in  the  direction  of  the  earth's  axis  (Arts.  205  and  214). 
Aberration  also  comes  in,  and  this  also  is  not  a  real  motion  of  the 
stars,  but  only  an  apparent  one. 

800.  Proper  Motions.  —  But  after  allowing  for  all  these  apparent 
and  common  motions,  which  depend  upon  the  stars'  places  in  the 
sky,  and  are  sensibly  the  same  for  all  stars  in  the  same  telescopic 


REAL   MOTIONS    OF    STAKS. 


493 


field  of  view,  whatever  may  be  their  real  distance  from  us,  we  find 
that  most  of  the  larger  stars  have  a  "-proper  motion  "  of  their  own 
("proper"  as  opposed  to  "common"),  which  displaces  them  slightly 
with  reference  to  the  stars  about  them.  There  are  only  a  few  stars 
for  which  this  proper  motion  amounts  to  as  much  as  1"  a  year ;  per- 
haps 150  such  stars  are  now  known,  but  the  number  is  constantly  in- 
creasing, as  more  and  more  of  the  smaller  stars  come  to  be  accurately 
observed. 

The  maximum  proper  motion  at  present  known  (detected  in  1898) 
is  that  of  the  8th  magnitude  star  No.  243  of  the  "  Fifth  hour  "  in 
the  Cordoba  Zone-Catalogue.  It  has  an  apparent  drift  of  8  ".7  annu- 
ally,—  enough  to  carry  it  360°  in  149000  years.  The  next  largest 
known  proper  motions  are  the  following :  — 


1830,  Groombridge, 

7th  mag. 

7".0 

e  Indi, 

5th  mag.  , 

4x/.5 

9352,  Lacaille, 

7th     " 

6".  9 

Lalande  21258 

8th     " 

4x/.4 

32416,  Gould, 

9th     " 

6".  2 

o2  Eridani 

6th     " 

4/x.4 

61  Cygni, 

6th     " 

5".  2 

fj,  Cassiopeise, 

5th     " 

3".  8 

Lalande  21185 

7th     " 

4".  7 

a  Centauri, 

1st     " 

3".  7 

The  proper  motions  of  Arcturus  (2".l),  and  of  Sirius  (1"!2), 
are  considered  "  large,"  but  are  exceeded  by  a  considerable  num- 
ber of  stars  besides  those  given  above.  Since  the  time  of  Ptolemy, 
Arcturus  has  moved  more  than  a  degree,  and  Sirius  about  half  as 
much.  These  motions  were  first  detected  by  Halley  in  1718. 

It  is  found,  as  might  be  expected,  that  the  brighter  stars,  which  as 
a  class  are  presumably  nearer  than  the  fainter  ones,  have  on  the 
average  a  greater  proper  motion ;  on  the  average  only,  however,  as 
is  evident  from  the  list  given  above.  Many  smaller  stars  have  larger 
proper  motions  than  any  bright  one,  for  there  are  more  of  them. 

The  average  proper  motion  of  the  first  magnitude  stars  is  about 
J"  annually,  and  that  of  the  sixth  magnitude  stars,  —  the  smallest 
visible  to  the  naked  eye,  —  is  about  ^". 


To  the  Earth 


801.  Real  Motions  of  Stars.  —  The  proper  motion  of  a  star  gives 
comparatively  little  information 
as  to  its  real  motion  until  we  know 
the  distance  of  the  star  and  the 
true  direction  of  the  motion,  since 
this  "proper  motion'7  as  deter- 
mined from  the  star-catalogues  is 
only  the  angular  value  of  that  part 
or  component  of  the  star's  whole  motion  which  is  perpendicular  to 


b  B 

FIG.  214. 

Components  of  a  Star's  Proper  Motion. 


494  THE   STARS. 

the  line  of  sight,  as  is  clear  from  the  figure.  When  the  star  really 
moves  from  A  to  B  (Fig.  214),  it  will  appear,  as  seen  from  the  earth, 
to  have  moved  from  A  to  b.  The  angular  value  of  Ab  as  seen  from 
the  earth  is  the  "  proper  motion  "  (usually  denoted  by  /*).  Expressed 
in  seconds  of  arc,  we  have 

(Ah      \ 
..       V 
distance/ 

A  body  moving  directly  towards  or  from  the  earth  has  no  "  proper 
motion  "  at  all,  speaking  technically,  —  none  that  can  be  obtained 
from  the  comparison  of  star-catalogues. 

0.          .,  .        .,         w."  X  distance 
Since  Ab  in  miles  =  ^  — > 

the  proper  motion  cannot  be  translated  into  miles  without  a  knowl- 
edge of  the  star's  distance,  and  at  present  we  know  the  distance  in 
only  a  very  few  cases  ;  nor  can  the  true  motion  AB  be  found  until 
we  also  know  either  the  angle  BAE  or  else  Aa,  the  "  motion  in  the 
line  of  sight "  or  "radial  motion,"  which,  as  we  shall  see  in  the  next 
article,  is  determined  by  spectroscopic  observations. 

But  since  AB  is  necessarily  greater  than  Ab,  it  is  possible  in  some  cases 
to  determine  a  minor  limit  of  velocity,  which  must  certainly  be  exceeded  by 
the  star.  In  the  case  of  1830,  Groombridge,  for  instance,  its  distance  is 
probably  about  1  500000  times  the  earth's  distance  from  the  sun.  Now 
at  that  distance  the  observed  proper  motion  of  7"  a  year  would  correspond 
to  an  actual  velocity1  along  the  line  Ab  of  about  150  miles  a  second,  and 
this  is  not  its  whole  motion.  According  to  Campbell  the  star  is  also  mov- 
ing towards  us  with  a  velocity  of  about  60  miles  a  second. 

In  the  case  of  61  Cygni  we  know  the  distance  to  be  just  about  500000 
times  that  of  the  earth  from  the  sun,  and  its  proper  motion  of  5".2  annually 
corresponds  therefore  to  a  distance  Ab  of  about  1200  million  miles,  and  a 
velocity  of  about  38  miles  a  second,  —  not  quite  twice  the  orbital  velocity 
of  the  earth.     We  shall  see  in  the  next  articles  how  the  velocity  A  a  can  be 
determined.     For  61  Cygni  it  has  recently  been  measured  by  Belopolsky 
and  is  found  to  be  about  34.5  miles  towards  us.     The  entire  velocity,  there- 
fore, along  AB  is  about  51  miles  (referred  to  the  sun  as  the  origin  of 
measures).     If  we  accept  the  parallax  of  the  star  C.  Z.,  V.,  243,  as  0".30, 
according  to  the  determination  of  Gill,  we  find  that  its  annual  drift  of  8".7 
corresponds  to  a  velocity  of  about  85  miles  a  second,  which  is  much  less 
than  that  of  1830  Groombridge. 

1  When  the  "parallax"  of  a  star  (Art.  808)  is  known,  this  "thwartwise 
velocity,"  or  "cross-motion,"  is  given  in  miles  per  second  by  the  formula 
©  =  2.944  x-jfj.  and  p  being  respectively  the  proper  motion  and  parallax. 


RADIAL   VELOCITY   OF    STAKS.  495 

802.    "Motion  in  the  Line  of  Sight"  or  " Radial  Velocity."1  — 

The  comparison  of  star-catalogues  furnishes  no  information  as  to 
the  motion  of  stars  towards  or  from  us,  but  when  a  star  is  bright 
enough  to  give  an  observable  spectrum  we  can  ascertain  the  rate  of 
its  approach  or  recession  (Aa  in  the  figure)  by  means  of  the  spectro- 
scope. If  its  distance  is  increasing,  then  (Art.  321)  its  lines  will 
be  shifted  towards  the  red,  and  towards  the  blue  if  it  is  coming 
nearer.  The  shift  is  ascertained  by  arranging  the  telespectroscope 
(Art.  313)  so  that  in  some  way,  by  a  "  comparison-prism  "  or  other- 
wise, the  observer  shall  have  close  together,  or  superposed,  the 
spectrum  of  the  star  he  is  dealing  with  and  also  that  of  some  sub- 
stance (say  hydrogen  or  iron)  whose  lines  are  present  in  the  star- 
spectrum  :  he  can  then  appreciate  and  measure  any  displacement  of 
the  stellar  lines.  Sir  William  Huggins  in  1867  was  the  first  to 
apply  this  method,  and  obtained  some  very  interesting  results,  quite 
sufficient  to  establish  its  feasibility,  although  from  the  insufficient 
power  of  his  instruments  they  can  now  be  regarded  only  as  first 
approximations.  The  work  has  since  been  followed  up  for  several 
years  at  Greenwich  and  some  other  places ;  but  so  long  as  visual 
observations  were  depended  upon  the  results  were  not  very  satis- 
factory. Observations  of  this  kind  are  extremely  difficult.  The 
star-spectra  are  faint  at  best,  the  displacements  of  the  lines  very 
minute,  and  the  lines  themselves  often  broad  and  hazy,  and  ill 
adapted  for  accurate  measurement ;  so  that  the  individual  results 
for  a  single  star  are  apt  to  be  mournfully  at  variance  with  each  other. 
In  the  case  of  the  nebulae,  however,  which  give  spectra  containing 
sharp  bright  lines,  the  Lick  observers  have  made  visual  observations 
which  fairly  compete  with  photographic  in  accuracy. 

802*.     Spectrographic  Determination  of  Radial  Velocity.  —  The 

unsatisfactory  results  of  visual  observations  led  Vogel  in  1888  to 
apply  photography,  and  with -great  success.  In  this  case  the  diffi- 
culties arising  from  the  faintness  of  the  star-spectra  can  be  largely 
overcome  by  prolonged  exposure,  and  the  necessary  measurements 
can  be  made  upon  the  negatives  at  leisure.  The  little  cut  (Fig.  215) 
shows  how  on  a  negative  of  the  spectrum  of  ft  Orionis  (Eigel)  the 
recession  of  the  star  is  shown  by  the  displacement  of  a  line  in  its 
spectrum  towards  the  red,  when  compared  with  the  corresponding 

1  We  shall  hereafter  follow  the  French  usage  in  employing  the  term  "  Eadial 
Velocity"  (Vitesse  Radiate)  to  denote  the  rate  at  which  a  body  is  changing  its 
distance  from  the  observer,  i.e.,  advancing  or  receding.  The  equivalent  expres- 
sion, «*  Motion  in  Line  of  Sight,"  is  rather  clumsy. 


496 


THE    STARS. 


Blue 


Red 


bright  line  (black  in  the  negative)  of  hydrogen.    Fig.  215*  (borrowed 
by  permission  from  Frost's  translation  of  Schemer's  Astronomical 

Spectroscopy)  shows  the  actual  appear- 
ance of  part  of  the  spectrum  of  a  Aurigse 
and  the  corresponding  part  of  the  solar 
spectrum,  as  seen  under  the  microscope 
with  which  the  measurements  are  made. 
The  solar  spectrum  is  of  course  on  a 
separate  plate,  but  this  plate  and  the 
star-negative  are  clamped  together  so  as  to  make  the  lines  corre- 
spond, and  facilitate  the  identification  of  the  star-lines.  The  sharp 
black  line  that  crosses  the  narrow  star-spectrum  is  the  Hydrogen  y 
line  in  the  spectrum  of  a  Geissler  tube  placed  in  the  cone  of  rays 
some  two  feet  above  the  slit-plate,  and  illuminated  by  electricity  for 


Spectrum  of  Rigel 

FIG.  215. 

Displacement  of  Hy  Line  in  the  Spec- 
trum (3  Orionis. 


FIG.  215*. 

a  few  seconds  at  various  times  during  the  long  exposure  (an  hour 
or  so)  which  is  required  for  the  star-spectrum.  One  sees  easily  that 
in  this  case  the  star-line  is  shifted  a  little  to  the  right ;  but  the  line 
appears  to  be  so  poorly  defined  that  accurate  measurement  would  be 
difficult ;  for  the  methods  by  which  the  difficulty  is  overcome,  and 
for  the  corrections  required  on  account  of  the  orbital  motion  of  the 
earth  and  other  causes,  the  reader  is  referred  to  the  book  from  which 
the  figure  is  taken.  It  is  found  that  the  probable  error  of  the 


FIG.  216 


498  THE    STABS. 

radial  velocity  of  a  star  as  deduced  from  the  Potsdam  photographs 
seldom  exceeds  a  mile  a  second,  and  is  usually  much  less. 

The  large  engraving,  Fig.  216,  is  from  a  photograph  of  the 
Potsdam  "  Spectrograph,"  as  the  photographic  spectroscope  is  called. 
It  is  taken  from  the  same  source  as  the  preceding  figure. 

Observations  made  by  Humphreys  and  Mohler  of  Baltimore  in  1895  appear 
to  show  that  under  heavy  pressures  the  spectrum-lines  of  many  elements 
shift  slightly  towards  the  red,  just  as  if  the  luminous  object  were  receding. 
The  shift  is  different  for  different  substances,  but  is  always  minute,  never, 
even  under  pressures  of  ten  or  twelve  atmospheres,  exceeding  the  displace- 
ment that  would  be  due  to  a  receding  velocity  of  one  or  two  miles  a  second. 
Still,  it  is  quite  sufficient  to  require  to  be  examined  and  taken  into  account 
in  all  applications  of  Doppler's  principle. 

Table  VII.  of  the  Appendix  presents  Vogel's  results  for  the  51  stars  that 
he  had  been  able  to  deal  with  up  to  1892.  His  telescope  (since  replaced  by 
a  much  larger  one)  had  an  aperture  of  only  eleven  inches,  which  limited 
him  to  the  brightest  stars.  The  maximum  velocity  indicated  by  his  obser- 
vations is  that  of  a  Tauri,  30.1  miles  a  second,  receding.  The  next  in  order 
is  that  of  y  Leonis,  24.1  miles,  approaching.  Belopolsky,  at  Pulkowa,  has 
since  found  for  £  Herculis  the  still  higher  velocity  of  43.8  miles,  —  also 
approaching.1  Investigations  of  this  nature  are  now  (1908)  successfully 
prosecuted  by  several  other  observers,  prominent  among  whom  are  Belo- 
polsky at  St.  Petersburg,  and  Campbell  and  Frost  at  the  Lick  and  Yerkes 
observatories.  The  number  of  stars  whose  radial  velocity  has  been  thus 
determined  is  about  200,  and  the  "probable  error  "has  been  reduced  to 
about  a  quarter  of  a  mile.  \ 

803.  Star-Groups.  —  Star-atlases  have  been  constructed  by  Proc- 
tor and  Flammarion,  which  show  by  arrows  the  direction  and  rate 
of  the  angular  proper  motion  of  the  stars  as  far  as  now  known.  A 


\ 

\ 
\. 


FIG.  217.  —  Common  Proper  Motions  of  Stars  in  the  "  Dipper  "  of  Ursa  Major. 

moment's  inspection  shows  that  in  many  cases  stars  in  the  same 
neighborhood  have  a  proper  motion  nearly  the  same  in  direction  and 
in  amount. 

1  Campbell  finds  for  T;  Cephei  a  radial  velocity  of  54  miles,  and  for  /t  Cassio- 
peise,  61,  both  approaching. 


THE    SUN'S   WAY.  499 

Thus,  Flammarion  has  pointed  out  that  the  stars  in  the  "  dipper  "  of  Ursa 
Major  have  such  a  community  of  motion,  except  a  and  17,  —  the  brighter  of 
the  pointers  and  the  star  in  the  end  of  the  handle,  —  which  are  moving  in 
entirely  different  directions,  and  refuse  to  be  counted  as  belonging  to  the 
same  group.  Fig.  217  shows  the  proper  motions  of  the  stars  which  compose 
this  group.  The  Potsdarn  observations  show  that  all  of  them,  excepting  8, 
which  is  too  faint  to  be  included,  are  approaching  the  sun  with  velocities 
between  12  and  20  miles  a  second. 

The  brighter  stars  of  the  Pleiades  are  found  in  the  same  way  to  have  a 
common  motion. 

In  fact,  it  appears  to  be  the  rule  rather  than  the  exception  that 
stars  apparently  near  each  other  are  really  connected  as  comrades, 
travelling  together  in  groups  of  twos  and  threes,  dozens  or  hundreds. 
They  show,  as  Miss  Clerke  graphically  expresses  it,  a  distinctly 
"  gregarious  tendency." 

804.  The  "Sun's  Way."  — The  proper  motions  of  the  stars  are 
due  partly  to  their  own  real  motion,  and  partly  also  to  the  motion  of 
our  sun,  which  is  moving  swiftly  through  space,  taking  with  it  the 
earth  and  the  planets.  Sir  William  Herschel  was  the  first  to  investi- 
gate and  determine  the  direction  of  this  motion  a  little  more  than  100 
years  ago.  The  principle  involved  is  this  :  that  the  apparent  motion 
of  each  star  is  made  up  of  its  own  motion  combined  with  the  motion 
of  the  sun  reversed  (Art.  492).  The  effect  must  be  that,  on  the  whole, 
the  stars  in  that  part  of  the  sky  towards  which  the  sun  is  moving  are 
separating  from  each  other, — the  intervals  between  them  widening  out, 
—  while  in  the  opposite  part  of  the  heavens  they  are  closing  up;  and 
in  the  intermediate  part  of  the  sky  the  general  drift  must  be  backward 
with  reference  to  the  sun's  (and  earth's)  real  motion.  Just  as  one 
walking  in  a  park  filled  with  people  moving  indiscriminately  in 
different  directions  would,  on  the  whole,  find  that  those  in  front  of 
him  appeared  to  grow  larger,1  and  the  spaces  between  them  to  open 
out,  while  at  the  sides  they  would  drift  backwards,  and  in  the  rear 
close  up. 

Spectroscopic  observations  ought  also  to  give  evidence  as  to  this 
solar  motion ;  and  they  do  so.  On  the  whole,  in  the  quarter  of  the 
heavens  towards  which  the  sun  is  moving,  the  stellar  spectra  indicate 
approach,  and  vice  versa  in  the  opposite  quarter.  As  yet,  however, 
the  stars  whose  radial  motion  is  known  are  too  few  to  yield  a  very 

1  Theoretically,  of  course,  the  stars  towards  which  we  are  moving  must  appear 
to  grow  brighter  as  well  as  to  drift  apart ;  but  this  change  of  brightness,  though 
real,  is  entirely  imperceptible  within  a  human  lifetime. 


500  THE   STARS. 

good  determination  of  the  direction  of  the  solar  motion,  though  they 
give  a  better  determination  of  its  velocity  than  is  furnished  by  the 
other  method. 

805.  About  twenty  different  determinations  of  the  point  in  the 
sky  towards  which  this  motion  of  the  sun  is  directed  have  been 
worked  out  by  various  astronomers,  using  in  their  discussions  the 
angular  proper  motions  of  from  twenty  to  twenty-five  hundred  stars. 
All  the  investigations  present  a  reasonable  accordance  of  results, 
and  show  that  the  sun  is  moving  towards  a  point  near  the  eastern  edge 
of  the  constellation  of  Hercules,  having  a  right  ascension  of  about  277°. 5 
and  a  declination  of  about  35°.     (According  to  Newcomb.) 

Several  of  the  later  results  deduced  from  the  proper  motions  of  the  fainter 
stars  indicate  a  point  farther  north  and  east,  near  a  Lyrae.  The  spectro- 
scopic  result,  on  the  contrary,  puts  it  farther  west  and  north ;  but  is  not 
entitled  to  much  weight. 

This  point  towards  which  the  sun's  motion  is  directed  is  called 
"The  Apex  of  the  Sun's  Way"lo?  often,  and  more  simply,  "  The 
Solar  Apex" 

806.  Velocity  of  the  Sun's  Motion  in  Space.  —  From  the  discus- 
sion of  the  "  proper  motions  "  it  appears  that  the  sun's  velocity  is 
such  as  would  carry  it,  and  its  system  with  it,  about  5"  in  a  century 
as  seen  from  the  average  sixth  magnitude  star,  the  sixth  magnitude 
being  the  smallest  easily  visible  to  the  naked  eye.     We  could  at 
once  convert  the  expression  into  miles  if  we  had  any  accurate  knowl- 
edge of  the  distance  of  this  class  of  stars,  but  as  to  that  we  can  do 
little  more  than  to  make  a  reasonable  conjecture.     If  we  accept  the 
estimate  of  Ludwig  Struve  (who  has  made  one  of  the  most  careful 
investigations  of  the  subject  of  "  proper  motions  "),  and  assume  this 
distance  to  be  about  20  000000  astronomical  units,  the  rate  of  the 
solar  motion  comes  out  about  fifteen  miles  a  second. 

The  spectroscopic  observations  of  "  motion  in  the  line  of  sight,"  on 
the  other  hand,  indicate  a  velocity  of  about  eleven  miles  a  second;  and 
although  these  data  are  not  yet  sufficient  to  furnish  a  determination 
that  can  be  considered  final,  this  value  is  probably  more  authoritative 
than  that  deduced  from  the  "proper  motions,"  because  it  is  not  in 
any  way  dependent  on  our  uncertain  knowledge  of  stellar  distances. 

Tt  is  to  be  noted  that  this  swift  motion  of  the  solar  system,  while  of 
course  it  affects  the  real  motion  of  the  planets  in  space,  converting  them 

1  See  note  on  page  506. 


THE   CENTRAL   SUN. 


501 


C\ 


into  a  sort  of  corkscrew  spiral  like  the  figure  (Fig.  218),  does  not  in  the 
least  affect  the  relative  motion  of  sun  and  planets,  as  some  paradoxers  have 
supposed  it  must. 

807.  The  Central  Sun.  —  We  mention  this  subject  simply  to  say 
that  there  is  no  real  foundation  for  the  belief  in  the  existence  of 
such  a  body.  The  idea  that  the 
motion  of  our  sun  and  of  the  other 
stars  is  a  revolution  around  some 
great  central  sun  is  a  very  fascinating 
one  to  certain  minds,  and  one  that 
has  been  frequently  suggested.  It 
was  seriously  advocated  some  fifty 
years  ago  by  Madler,  who  placed  this 
centre  of  the  stellar  universe  at 
Alcyone,  the  principal  star  in  the 
Pleiades. 

It  is  certainly  within  bounds  to 
deny  that  any  such  motion  has  been 
demonstrated,  and  it  is  still  less 
probable  that  the  star  Alcyone  is 
the  centre  of  such  a  motion,  if  the 
motion  exists.  So  far  as  we  can  FIG.  218. 

-judge    at    present    it    is    most    likely    The  Earth's  Motion  in  Space  as  affected 
1,     '      ,  v  .  ,     .  by  the  Sun's  Drift. 

that  the   stars  are   moving,  not  m 

regular  closed  orbits  around  any  centre  whatever,  but  rather  as 
bees  do  in  a  swarm,  each  for  itself,  under  the  action  of  the  predomi- 
nant attraction  of  its  nearest  neighbors.  The  solar  system  is  an 
absolute  monarchy  with  the  sun  supreme.  The  great  stellar  system 
appears  to  be  a  republic,  without  any  such  central,  unique,  and 
dominant  authority. 


THE  PARALLAX  AND  DISTANCE  OF  THE  STARS. 

808.  When  we  speak  of  the  "parallax  "  of  the  moon,  the  sun,  or 
a  planet,  we  always  mean  the  diurnal  parallax,  i.e.,  the  angular 
semi-diameter  of  the  earth  as  seen  from  the  body  in  question.  In 
the  case  of  the  stars,  this  kind  of  parallax  is  hopelessly  insensible, 
never  reaching  an  amount  of  ^J^  of  a  second  of  arc. 

The  expression  "  parallax  of  a  star  "  always  means  its  annual  par- 
allax, that  is,  the  semi-diameter  of  the  earth's  orbit  as  seen  from  the 
star.  Even  this  in  the  case  of  all  stars  but  a  very  few  is  a  mere  frac- 


502  THE    STARS. 

tion  of  a  second  of  arc,  too  small  to  be  measured.  In  a  few  instances 
it  rises  to  about  half  a  second,  and  in  the  one  case  of  our  nearest 
neighbor  (so  far  as  known  at  present),  the  star  a  Centauri,  it  appears 
to  be  about  0".9,  according  to  the  earlier  observers,  or  about  0".75 
according  to  the  latest  determination  of  Gill  and  Elkin.  In  Fig.  219 
the  angle  at  the  star  is  the  star's  parallax. 

In  accordance  with  the  principle  of  relative  motion  (Art.  492), 
every  star  has,  superposed  upon  its  own  motion  and  combined  with 
it,  an  apparent  motion  equal  to  that  of  the  earth  but  reversed.  If 
the  star  is  really  at  rest  it  must  seem  to  travel  around  each  year  in  a 
little  orbit  186  000000  miles  in  diameter,  the  precise  counterpart  of 
the  earth's  orbit  in  size  and  form,  and  having  its  plane  parallel  to 
the  ecliptic. 


Star 


FIG.  219.  —  The  Annual  Parallax  of  a  Star. 

If  the  star  is  near  the  pole  of  the  ecliptic  this  apparent  "  parallactic  "  orbit 
will  be  viewed  perpendicularly  and  appear  as  a  circle ;  if  the  star  is  on  the 
ecliptic  it  will  be  seen  edgewise  as  a  short,  straight  line,  while  in  the  inter- 
mediate latitudes  the  parallactic  orbit  will  appear  as  an  ellipse.  In  this 
respect  it  is  just  like  the  "aberrational"  orbit  of  a  star  (Art.  226);  but 
while  the  aberrational  orbit  is  of  the  same  size  for  every  star,  having  always 
a  semi-major  axis  of  20".47,  the  size  of  the  parallactic  orbit  depends  upon 
the  distance  of  the  star.  Moreover,  in  the  parallactic  orbit  the  star  is 
always  opposite  to  the  earth,  while  in  the  aberrational  orbit  it  keeps  just 
90°  ahead  of  her. 

809,  If  we  can  find  a  way  of  measuring  this  parallactic  orbit,  the 
star's  distance  is  at  once  determined  from  the  equation 

206265  X  R 

Distance  = r. > 

p" 

in  which  p"  is  the  parallax  in  seconds  of  arc  (the  apparent  semi- 
major  axis  of  the  parallactic  orbit),  and  H  is  the  earth's  distance 
from  the  sun. 

The  determination  of  stellar  parallax  had  been  attempted  over  and 
over  again  from  the  days  of  Tycho  down,  but  without  success  until 
Bessel,  in  1838,  succeeded  in  demonstrating  and  measuring  the  par- 
allax of  the  star  61  Cygni ;  and  the  next  year  Henderson,  of  the 


METHODS   OF   DETERMINING   STELLAR   PARALLAX.        503 

Cape  of  Good  Hope,  brought  out  that  of  a  Centauri.  It  will  be 
remembered  that  it  was  mainly  on  account  of  his  failure  to  detect 
stellar  parallax  that  Tycho  rejected  the  Copernican  theory  and 
substituted  his  own  (Art.  504). 

Roemer,  of  Copenhagen,  in  1690,  thought  that  he  had  detected  the  effect 
of  stellar  parallax  in  his  observations  of  the  difference  of  right  ascension 
between  Sirius  and  Vega  at  different  times  of  the  year.  A  few  years  later, 
Horrebow,  his  successor,  from  his  own  discussion  of  Roemer's  observations, 
made  out  the  amount  to  be  nearly  four  seconds  of  time  or  I/,  and  published 
his  premature  exultation  in  a  book  entitled  "Copernicus  Triumphans." 
The  discovery  of  aberration  by  Bradley  explained  many  abnormal  results  of 
the  early  astronomers  which  had  been  thought  to  arise  from  stellar  parallax, 
and  proved  that  the  parallax  must  be  extremely  small.  About  the  begin- 
ning of  the  present  century,  Brinkley,  of  Dublin,  and  Pond,  the  Astronomer 
Royal,  had  a  lively  controversy  over  their  observations  of  a  Lyrae  (Vega). 
Brinkley  considered  that  his  observations  indicated  a  parallax  of  nearly  3". 
Pond,  on  the  other  hand,  deduced  a  minute  negative  parallax,  which  is  of 
course  impossible,  and,  as  some  one  expresses  it,  would  put  the  star  "  some- 
where on  the  other  side  of  nowhere."  In  fact,  as  it  turns  out,  Pond  was 
nearer  right  than  Brinkley,  the  actual  parallax  as  deduced  from  the  latest 
observations  being  only  about  0".2.  The  apparent  negative  parallax  simply 
indicated  that  the  actual  parallax  is  too  small  to  show  itself  decidedly,  and 
was  overborne  by  errors  of  observation.  The  periodical  changes  of  tempera- 
ture and  air  pressure  continually  lead  to  fallacious  results  except  under  the 
most  extreme  precautions. 

810.  Methods  of  Determining  Parallax, — The  operation  of  meas- 
uring a  stellar  parallax  is,  on  the  whole,  the  most  delicate  in  the 
whole  range  of  practical  astronomy.  Two1  methods  have  been  suc- 
cessfully employed  so  far  —  the  absolute  and  the  differential. 

(a)  The  first  method  consists  in  making  meridian  observations  of 
the  right  ascension  and  declination  of  the  star  in  question  at  differ- 
ent seasons  of  the  year,  applying  all  known  corrections  for  preces- 
sion, nutation,  aberration,  and  proper  motion,  and  then  studying  the 
resulting  star-places.  If  the  star  is  without  parallax,  the  places 
should  be  identical  after  the  corrections  have  been  duly  applied.  If 
it  has  parallax,  the  star  will  be  found  to  change  its  right  ascension 
and  declination  systematically,  though  slightly,  through  the  year.  But 
the  changes  of  the  seasons  so  disturb  the  constants  of  the  instrument 
that  the  method  is  treacherous  and  uncertain.  There  is  no  possibil- 
ity of  getting  rid  of  these  temperature  effects  (in  producing  changes 
of  refraction  and  varying  expansions  of  the  instrument  itself)  by 
merely  multiplying  observations  and  taking  averages,  since  the 

1  For  spectroscopic  method,  see  note  on  page  506. 


504  THE   STARS. 

changes  of  temperature  are  themselves  annually  periodic,  just  as  is 
the  parallax  itself. 

Still,  in  a  few  cases  the  method  has  proved  successful.  Different 
observers  at  different  places  with  different  instruments  have  found 
for  a  few  stars  fairly  accordant  results;  as,  for  instance,  in  the  case 
of  a  Centauri,  already  mentioned  as  our  nearest  neighbor. 

811.  (b)   The  Differential  Method.  —  This  consists  in  measuring 
the  change  of  position  of  the  star  whose  parallax  we  are  seeking 
(which  is  supposed  to  be  comparatively  near  to  us),  with  reference 
to  other  small  stars,  which  are  in  the  same  telescopic  field  of  view, 
but  are  supposed  to  be  so  far  beyond  the  principal  star  as  to  have 
no  sensible  parallax  of  their  own.     If  the  comparison  stars  are  near 
the  large  one  (say  within  two  or  three  minutes  of  arc),  the  ordinary 
wire  micrometer  answers  very  well  for  the  necessary  measures  ;  but 
if  they  are  farther  away,  the  heliometer  (Art.  677)  represents  special 
and  very  great  advantages.     It  was  with  this  instrument  that  Bessel, 
in  the  case  of  61  Cygni,  obtained  the  first  success  in  this  line  of 
research. 

The  great  advantage  of  the  differential  method  is  that  it  avoids 
entirely  the  difficulties  which  arise  from  the  uncertainties  as  to  the 
exact  amount  of  precession,  etc. ;  and  in  great  measure,  though  not 
entirely,  those  arising  from  the  effect  of  the  seasons  upon  refraction 
and  the  condition  of  the  instruments.  On  the  other  hand,  however, 
it  gives  as  the  final  result,  not  the  absolute  parallax  of  the  star,  but 
only  the  difference  between  its  parallax  and  that  of  the  comparison 
star.  If  the  work  is  accurate  the  parallax  deduced  cannot  be  too 
great;  but  it  may  be  sensibly  too  small,  and  so  may  make  the  star 
apparently  too  remote.  This  is  because  the  parallax  of  the  com- 
parison star  can  never  be  quite  zero :  if  the  comparison  star  happens 
to  have  a  parallax  of  its  own  as  large  as  that  of  the  principal  star, 
there  will  be  no  relative  parallax  at  all ;  if  larger,  the  parallax  sought 
will  come  out  apparently  negative,  which  is  by  no  means  unusual. 

812.  Determination  of  Parallax  by  Photography.  —  Eecently 
photography  has  been  pressed  into  the  service  with  great  advantage, 
first  by  the  late  Professor  Pritchard  in  1886.     Measurements  with 
the  micrometer  and  heliometer  are  so  tedious  that  in  practice  it  is 
impossible  to  use  more  than  one  or  two  "comparison  stars"  in 
determining  a  parallax  by  their  use,  but  there  is  no  such  restriction 
in  the  case  of  photography.     The  negative  will  probably  show  the 
images  of  a  great  number  of  stars,  and  all  of  them  can  be  utilized. 


SELECTION   OF    STARS.  505 

Of  course  the  number  of  negatives  must  be  considerable,  and  taken 
at  times  well  distributed  through  the  year,  with  extreme  precautions 
also  in  the  development  of  the  plates  to  prevent  any  slipping  or 
distortion  of  the  film  upon  the  glass. 

It  is  probable  that  this  method  will  now  give  us  before  very  long 
a  large  number  of  well-determined  parallaxes. 

813.  Selection  of  Stars.  —  It  is  important  to  select  for  investigations 
of  this  kind  those   stars  which  may  reasonably  be  supposed  to  be  near, 
and,  therefore,  to  have  a  sensible  parallax.     The  most  important  indication 
of  proximity  is  a  large  proper  motion,  and  brightness  is,  of  course,  confirm- 
atory.    At  the  same  time,  while  it  is  probable  that  a  bright  star  with  large 
proper  motion  is  comparatively  near,  it  is  not  certain.      The  small  stars 
are  so  much  more  numerous  than  the  large  ones  that  it  will  be  nothing  sur- 
prising if  we  should  find  among  them  one  or  more  neighbors  nearer  than 
a  Centauri  itself. 

814.  Unit  of  Stellar  Distance.—  The  Light-  Year.     The  ordinary 
"  astronomical  unit"  or  distance  of  the  sun  from  the  earth,  is  not 
sufficiently  large  to  be  convenient  in  expressing  the  distances  of  the 
stars.     It  is  found  more  satisfactory  to  take  as  a  unit  the  distance 
that  light  travels  iri  a  year,  which  is  about  63000  times  the  distance 
of  the  earth  from  the  sun.     A  star  with  a  parallax  of  1"  is  at  a  dis- 
tance of  3.26  "  light-years"  so  that  the  distance  of  any  star  in  "light- 
years"  is  expressed  by  the  formula 


815.  Table  IV.  in  the  Appendix  gives  the  parallaxes,  the  dis- 
stances  in  light-years,  and  the  proper  motions,  and  "cross,"  or 
"  thwartwise  "  motions  (Art.  801,  note)  of  certain  stars  whose  par- 
allaxes may  be  considered  as  now  fairly  determined.  There  are 
other  stars  for  which  parallaxes  have  been  found  larger  than  some 
of  those  included  in  the  table  ;  but  the  results  are  not  yet  sufficiently 
confirmed. 

The  student  will,  of  course,  see  that  the  tabulated  distance  in  the  case  of 
a  remote  star  is  liable  to  an  enormous  percentage  of  error.  Considering  the 
amount  of  discordance  between  the  results  of  different  observers,  it  is 
extremely  charitable  to  assume  that  any  of  the  parallaxes  are  certain 
within  -^j  of  a  second  of  arc  :  but  in  the  case  of  a  star  like  the  pole  star, 
which  appears  to  have  a  parallax  of  less  than  0".08,  this  -^  of  a  second 
is  ^  of  the  whole  amount  ;  so  that  the  distance  of  that  star  is  uncertain  by 
at  least  twenty-five  per  cent.  (^L  of  a  second  is  the  angle  subtended  by  T^ 
of  an  inch  at  the  distance  of  ten  miles.) 


506  EXERCISES    ON   CHAPTER   XX. 

As  regards  the  distance  of  stars,  the  parallax  of  which  has  not  yet  been 
measured,  very  little  can  be  said  with  certainty.  It  is  probable  that  the 
remoter  ones  are  so  far  away  that  light  in  making  its  journey  occupies  a 
thousand  and  perhaps  many  thousand  years. 


EXERCISES  ON  CHAPTER  XX. 

1.  Assuming  the  parallax  of  61  Cygni  as  0".40,  and  that  it  is  approach- 
ing the  sun  at  the  rate  of  34.5  miles  a  second  (Art.  801),  how  many  years 
would  it  take  to  increase  its  brightness  by  ten  per  cent,  supposing  its  radial 
velocity  to  remain  unchanged?  ^  2050  yearg 


2.  Assuming  the  distance  of  61  Cygni  as  8.15  light-years,  and  that  the 
radial  and  transverse  velocities  are  34.5  and  38  miles  a  second  respectively, 
find  how  near  the  star  will  come  to  the  sun  if  it  keeps  on  uniformly,  and 
in  a  straight  line  ;  also  how  long  it  will  take  to  reach  that  point  of  nearest 
approach. 

Ans.   Nearest  approach  6.03  light-years  ;  reached  after  19900  years. 

NOTE  ON  SPECTROSCOPIC  DETERMINATION  OF  PARALLAX  OF  BINARY  STARS. 

SPECTEOSCOPIC  DETERMINATION  OF  STELLAR  PARALLAX.  In  the  case 
of  certain  binary  stars  (Arts.  872-877)  of  which  the  period  and  angular  dimen- 
sions of  the  orbit  are  accurately  known,  and  which  have  the  plane  of  the  orbit 
nearly  directed  toward  the  sun,  the  spectroscope  will  enable  us  to  determine  the 
velocity  with  which  they  move,  and  therefore  the  actual  dimensions  of  the  orbit 
in  miles.  Combining  this  with  the  apparent  dimensions  of  the  orbit  in  seconds 
of  arc,  we  get  at  once  the  length  in  miles  of  a  second  of  arc  at  the  star's  dis- 
tance ;  from  which  the  parallax  and  distance  immediately  follow. 

Wright  has  recently  (1905)  tested  this  method  upon  Alpha  Centauri,  and 
gets  a  parallax  of  0.76",  in  almost  absolute  agreement  with  that  determined  by 
the  usual  methods.  It  will,  however,  probably  be  a  long  time  before  we  shall 
have  sufficient  spectroscopic  observations  to  enable  us  to  make  many  applications 
of  the  method,  especially  as  the  available  binaries  are  not  numerous. 

NOTE  TO  ART.  805. 

Recent  investigations  of  Kapteyn  and  others  would  indicate  that  the  stars  are 
not  moving  indiscriminately  in  all  directions,  but  that  there  are  extensive  sys- 
tematic drifts.  The  nearer  stars  seem  to  belong  mainly  to  two  great  systems, 
streaming  in  opposite  directions.  This  being  the  case,  the  computed  position  of 
the  "  Solar  Apex"  will  be  affected. 


THE   LIGHT   OF   THE   STARS.  507 


CHAPTER    XXI. 

THE  LIGHT  OF  THE  STARS. —  STAR  MAGNITUDES  AND  PHO- 
TOMETRY.—  VARIABLE  STARS. STELLAR  SPECTRA.  —  SCIN- 
TILLATION OF  STARS. 

816.  Star  Magnitudes.  —  The  term  "magnitude,"  as  applied  to 
a  star,  refers  simply  to  its  brightness.     It  has  nothing  to  do  with 
its  apparent  angular  diameter.    Hipparchus  and  Ptolemy  arbitrarily 
graded  the  visible  stars,   according  to  their   brightness,   into  six 
classes,  the  stars  of  the  sixth  magnitude  being  the  smallest  visible 
to  the  eye,  while  the  first  class  comprises   about  twenty  of  the 
brightest.     There  is  no  assignable  reason  why  six  classes  should 
have  been  constituted,  rather  than  eight  or  ten. 

After  the  invention  of  the  telescope  the  same  system  was  extended 
to  the  smaller  stars,  but  without  any  general  agreement  or  concert, 
so  that  the  magnitudes  assigned  by  different  observers  to  telescopic 
stars  vary  enormously.  Sir  William  Herschel,  especially,  used  very 
high  numbers:  his  twentieth  magnitude  being  about  the  same  as 
the  fourteenth  on  the  scale  now  generally  used,  which  more  nearly 
corresponds  with  that  of  Argelander. 

817,  Fractional  Magnitudes.  —  Of  course,  the  stars  classed  to- 
gether under  one  magnitude  are  not  exactly  alike  in  brightness,  but 
shade  from  the  brighter  to  the  fainter,  so  that  exactness  requires  the 
use  of  fractional  magnitudes.     It  is  now  usual  to  employ  decimals 
giving  the  brightness  of  a  star  to  the  nearest  tenth  of  a  magnitude. 
Thus,  a  star  of  4.3  magnitude  is  a  shade  brighter  than  one  of  4.4, 
and  so  on. 

A  peculiar  notation  was  employed  by  Ptolemy,  and  used  by  Argelander 
in  his  "  Uranometria *  Nova."  It  recognizes  thirds  of  a  magnitude  as  the 
smallest  subdivision.  Thus,  2,  2,3,  3,2,  and  3  express  the  gradations 
between  second  and  third  magnitude,  2,3  being  applied  to  a  star  whose 
brightness  is  a  little  inferior  to  the  second,  and  3,2  to  one  a  little  brighter 
than  the  third  magnitude. 



1  The  term  "  Uranometria  "  has  come  to  mean  a  catalogue  of  naked-eye  stars  i 
like  the  catalogues  of  Hipparchus,  Ptolemy,  and  Ulugh  Beigh. 


508  STARS    VISIBLE    TO    THE    NAKED    EYE. 

818,  Stars  Visible  to  the  Naked  Eye.  —  Heis  enumerates  the 
stars  clearly  visible  to  the  naked  eye  in  the  part  of  the  sky  north 
of  35°  south  declination,  as  follows :  — 


1st  magnitude, 14 

2d  "  48 

3d  "  ,    152 


4th  magnitude, 313 

5th          »  854 

6th          "  .  2010 


Total 3391 

According  to  Newcomb,  the  number  of  stars  of  each  magnitude  is  such 
that  united  they  would  give,  roughly  speaking,  somewhere  nearly  the  same 
amount  of  light  as  that  received  from  the  aggregate  of  those  of  the  next 
brighter  magnitude.  But  the  relation  is  very  far  from  exact,  and  seems  to 
fail  entirely  for  the  fainter  magnitudes  below  the  tenth  or  eleventh,  the 
smaller  stars  being  less  numerous  than  they  should  be.  In  fact,  if  the  law 
held  out  perfectly,  and  if  light  was  transmitted  through  space  without  loss, 
the  whole  sky  would  be  a  blaze  of  light  like  the  surface  of  the  sun. 

819,  Light-Ratio  and  Absolute  Scale  of  Star  Magnitudes.  —  It 

was  found  by  Sir  John  Herschel,  about  fifty  years  ago,  that  the  light 
given  by  the  average  star  of  the  first  magnitude  is  just  about  one 
hundred  times  as  great  as  that  received  from  one  of  the  sixth,  and 
that  a  corresponding  ratio  has  been  pretty  nearly  maintained 
throughout  the  scale  of  magnitudes,  the  stars  of  each  magnitude 
being  about  2-J-  times  (  ^/IQO)  brighter  than  those  of  the  next  infe- 
rior magnitude.  The  number  which  expresses  the  ratio  of  the  light 
of  a  star  to  that  of  another  one  magnitude  fainter  is  called  the  light- 
ratio. 

In  the  star  magnitudes  of  the  maps  by  Argelander,  Heis,  and 
others,  which  are  most  used  at  present,  the  divergence  from  a  strict 
uniformity  of  light-ratio  is,  however,  sometimes  very  serious.  About 
1850  it  was  proposed  by  Pogson  to  reform  the  system,  by  adopting 
a  scale  with  the  uniform  light-ratio  of  -\yioO,  adjusting  the  first  six 
magnitudes  to  correspond  as  nearly  as  possible  with  the  magnitudes 
hitherto  assigned  by  leading  authorities,  and  then  carrying  forward 
the  scale  indefinitely  among  the  telescopic  stars.  Until  recently  this 
"  absolute  scale  of  magnitude"  as  it  has  been  called,  has  not  been 
much  used;  but  in  the  New  Uranometrias  made  at  Cambridge  (U.  S.) 
and  Oxford  it  has  been  adopted,  and  it  is  now  rapidly  supplanting 
the  older  systems. 

820.  Relative  Brightness  of  Different  Star  Magnitudes.  — In  this 
scale  the  light-ratio  between  successive  magnitudes  is  made  exactly 


NEGATIVE   MAGNITUDES.  509 

V'lOO?  or  the  number  whose  logarithm  is  0.4000, *  viz.,  2.512.  Its 
reciprocal  is  the  number  whose  logarithm  is  9.6000,  viz.,  0.3981.  If 
bl  is  the  brightness  of  a  standard  first-magnitude  star,  expressed 
either  in  candle-power  or  other  convenient  unit,  and  bn  be  the 
brightness  of  a  star  of  the  n\h  magnitude  on  this  scale,  we  shall 
therefore  have 

log  bn  =  log  bm—  T%  (n  —  m);  (1) 

conversely,  n  =  1  -f-  Y  (log  l>m  —  log  &„),  or  1  -f  -y>  log  lr 

\    n 

(n  —  m)  in  these  equations  being  the  number  of  magnitudes  between 
the  star  of  the  wth  magnitude  and  the  star  of  the  nth  magnitude ; 
so  that  for  a  star  of  the  sixth  magnitude  compared  with  the  first 
equation  (1)  reads, 

log  ^6  =  log  b,  -  T%  X  5  =  log  b,-2. 

With  this  light-ratio,  every  difference  of  five  magnitudes  corresponds 
to  a  multiplication  or  division  of  the  star's  light  by  100;  i.e.,  to  make 
one  star  as  bright  as  the  standard  star  of  the  first  magnitude  it  would 
require  100  of  the  sixth,  10000  of  the  eleventh,  1000000  of  the 
sixteenth,  and  100  000000  of  the  twenty-first  magnitude. 

As  nearly  standard  stars  of  the  first  magnitude  on  this  scale  we 
have  a  Aquilse  and  Aldebaran  (a  Tauri).  The  pole  star  and  the  two 
"  pointers  "  are  very  nearly  standard  stars  of  the  second  magnitude. 

821.  Negative  Magnitudes.  —  According  to  this  scale,  stars  that  are 
one  magnitude  brighter  than  those  of  the  standard  first  would  be  of  the  zero 
magnitude  (as  is  the  case  with  Arcturus),  and  those  that  are  brighter  yet 
would  be  of  a  negative  magnitude  ;  e.g.,  the  magnitude  of  Sirius  is  —  1.43  ; 
and  Jupiter  at  opposition,  in  conformity  to  this  system,  is  described  as  a 
star  of  nearly  —  2d  magnitude,  which  means  that  it  is  nearly  2.513,  or  about 
16  times  brighter  than  a  star  of  the  +lst  magnitude  like  Aldebaran. 

822.  Relation  of   Size  of  Telescope  to  the   Magnitude   of   the 
Smallest  Star  Visible  with  it.  —  If  a  telescope  just  shows  a  star  of 
a  given  magnitude,  then  to  show  stars  one  magnitude  smaller  we 
require  an  instrument  having  its   aperture  larger  in  the  ratio  of 

^/2.512  (or  VlOO)  ^°  1>  ^-e">  as  1-59  '•  !•  Every  tenfold  increase  in 
the  diameter  of  the  object-glass  will  therefore  carry  the  power  of 
vision  just  five  magnitudes  lower. 


510 


MEASUREMENT    OF    STAR   MAGNITUDES. 


Assuming  what  seems  to  be  very  nearly  true  for  normal  eyes  and  good 
telescopes,  that  the  "  minimum  visibile  "  for  a  one-inch  aperture  is  a  star  of 
the  ninth  magnitude,  we  obtain  the  following  little  table  of  apertures  required 
to  show  stars  of  a  given  magnitude,  the  formula  being  m  =  9  +  5  X  log.  of 
aperture  in  inches. 


Star  Magnitude  .  . 
Aperture  . 

7 
Oin.40 

8 
Oin63 

9 
lin  00 

10 

lin  59 

11 

2in  51 

12 

Sin  Q« 

Star  Magnitude  .  . 
Aperture  .  . 

13 

6in  31 

14 

10in  00 

15 

15in  90 

16 
25in  10 

17 

39in  3Q 

18 
fiHin  10 

But  on  account  of  the  increased  thickness  necessary  in  the  lenses  of  large 
telescopes,  they  never  quite  equal  their  theoretical  capacity  as  compared 
with  smaller  ones. 

The  Yerkes  telescope,  therefore  (forty  inches  aperture),  will  barely  show 
stars  of  the  seventeenth  magnitude,  not  quite  one  magnitude  fainter  than 
the  smallest  visible  with  the  twenty-six-inch  telescope  at  Washington.  But 
the  number  made  visible  will  probably  be  about  doubled,  since  the  smaller 
stars  are  vastly  the  more  numerous. 

823.  Measurement  of  Star  Magnitudes  and  Brightness.  —  Until 
recently  all  such  measurements  were  mere  eye-estimates.    Even  yet 
nearly  all  photometric  measures  depend  ultimately  on  the  judgment 
of  the  eye.    But  it  is  possible  by  the  help  of  instruments  to  aid  this 
judgment  very  much  by  limiting  the  point  to  be  decided,  to  the 
question  whether  two  lights  as  seen  are,  or  are  not,  exactly  equal, 
or  else  making  the  decision  depend  on  the  visibility  or  non-visibility 
of  some  appearance.     See  Art.  831. 

824.  Method  of  Sequences.  —  For  some  purposes  the  unassisted 
eye  is   quite  as  good  as  any  photometric  instrument.     It  judges 
directly  with  great  precision  of  the  order  of  brightness  in  which  a 
number  of  objects  stand.     In  the  method  of  "  sequences,"  as  it  is 
called,  the  observer  merely  arranges  the  stars  he  is  comparing,  say 
to  the  number  of  fifty  or  so,  in  the  order  of  their  brightness,  taking 
care  that  the  stars  in  each  sequence  list  are  nearly  at  the  same  alti- 
tude, and  seen  under  equally  favorable  circumstances.     Then  he 
makes  a  second  sequence,  taking  care  to  include  in  it  some  of  the 
stars  that  were  in  the  first;  and  so  on.     Finally,  from  the  whole  set 
of  sequences,  a  list  can  be  formed,  including  all  the  stars  contained 
in  any  of  them,  arranged  in  the  order  of  brightness.     This  process 
gives,   however,  no  determination  of  the   light-ratio,  nor   of  the 


INSTRUMENTAL   METHODS.  511 

number  of  times  by  which  the  light  of  the  brightest  exceeds  that 
of  the  faintest. 

Variable  stars  are  still  often  observed  in  this  way,  the  stars  with  which 
they  are  compared  being  such  as  have  their  magnitudes  already  well  deter- 
mined. 

825.  2.     Instrumental  Methods.  —  These  are  nearly  all  based  on 
two  different  principles:  — 

a.  The  measurement  is  made  by  causing  the  star  to  disappear  by 
diminishing   its    light  in  some  measurable  way.     This  is  usually 
referred  to  as  the  " method  of  extinctions" 

b.  The  measurement  is  effected  by  causing  the  light  of  the  star 
to  appear  just  equal  to  some  other  standard  light,  by  decreasing  the 
brightness  of  the  star  or  of  the  standard  in  some  known  ratio  until 
they  are  perfectly  equalized. 

Under  the  first  head  come  the  photometers  which  act  upon  the  principle 
of  "  limiting  apertures"  The  telescope  is  fitted  with  some  arrangement,  often 
a  so-called  "  cat's-eye,"  by  which  the  available  aperture  of  the  object-glass 
can  be  diminished  at  will,  and  the  observation  consists  in  determining  with 
what  area  of  object-glass  the  star  is  just  visible.  The  method  is  embar- 
rassed by  constant  errors  from  the  fact  that  the  greater  thickness  of  the 
glass  in  the  middle  of  the  lens  comes  into  account,  and,  still  worse,  from 
the  fact  that  the  image  of  the  star  becomes  large  and  diffuse  (on  account 
of  diffraction)  when  the  aperture  is  very  much  reduced. 

826.  The  Wedge  Photometer.  —  The  method  of  producing  the 
"  extinction  "  by  a  "  wedge  "  of  dark,  neutral-tinted  glass  is  much 
better.     The  wedge  is  usually  five  or  six  inches  long,  by  perhaps  a 
quarter  of  an  inch  wide,  and  at  the  thick  end  cuts  off  light  enough 
to  extinguish  the  brightest  stars  that  are  to  be  observed.     In  the 
Pritchard  form  of  the  instrument  this  wedge  is  placed  close  to  the 
eye  at  the  eye-hole  of  the  eye-piece ;  in  some  other  forms  it  is  placed 
at  the  principal  focus  of  the  object-glass,  where  micrometer  wires 
would  be  put.     . 

In  observation  the  wedge  is  pushed  along  promptly  until  the  star 
just  disappears,  and  a  graduation  on  the  edge  of  the  slider  is  read. 

The  great  simplicity  of  the  instrument  commends  it,  and  if  the  wedge  is 
a  good  one  of  really  neutral  glass  (which  is  not  easy  to  get),  the  results  are 
remarkably  accurate.  But  the  observations  are  very  trying  to  the  eyes  on 
account  of  the  straining  to  keep  in  sight  an  object  just  as  it  is  becoming 
invisible.  The  constant  of  the  wedge  must  be  carefully  determined  in  the 


512  STELLAR   PHOTOMETRY. 

laboratory,  i.e.,  what  length  of  the  wedge  corresponds  to  a  diminution  of 
the  light  of  a  star  by  just  one  magnitude  (cutting  off  0.602  of  its  light). 
It  is  convenient  to  have  the  slider  graduated  into  inches  or  millimeters  on 
the  one  edge  and  magnitudes  on  the  other.  The  "  Uranometria  Nova  Oxo- 
niensis"  is  a  catalogue  of  the  magnitudes  of  the  naked-eye  stars  to  the 
number  of  2784,  between  the  pole  and  10°  south  declination,  observed  with 
an  instrument  of  this  kind  by  Professor  Pritchard,  and  published  in  1885. 

827.  Polarization  Photometers. — The  instruments,  however,  with 
which  most  of  the  accurate  photometric  work  upon  the  stars  has 
been  done,  are  such  as  compare  the  light  of  the  star  with  some 
standard   by  means   of   an   "  equalizing  apparatus "  based  on  the 
application  of  the  principles  of  double  refraction  and  polarization. 

The  light  of  either  the  observed  star  or  the  comparison  star  (real  or  arti- 
ficial) is  polarized  by  transmission  through  a  Nicol  prism,  or  else  both  pen- 
cils are  sent  through  a  double  refracting  prism.  The  images  are  viewed 
with  a  Nicol  prism  in  the  eye-piece;  and  by  turning  this  the  polarized 
image  or  images  can  be  varied  in  brightness  at  pleasure,  and  the  amount  of 
variation  determined  by  reading  a  small  circle  attached  to  it.  In  the  pho- 
tometers of  Seidel  and  Zollner,  who  observed  comparatively  few  objects,  but 
very  accurately,  the  artificial  star  with  which  the  real  stars  were  compared 
was  formed  by  light  from  a  petroleum  lamp,  shining  through  a  small 
aperture,  and  reflected  to  the  eye  by  a  plate  of  glass  in  the  telescope  tube. 
Miiller  and  Kempf  at  Potsdam  used  the  same  instrument  in  their  very 
accurate  photometric  catalogue  of  3500  stars  above  the  7£  magnitude 
between  the  equator  and  dec. +20°.  Pickering,  in  making  his  much  more 
extensive  but  less  precise  photometric  catalogues,  published  and  in  progress, 
has  also  used  the  polarization  principle,  but  has  made  the  pole  star  the 
standard,  bringing  it  by  an  ingenious  arrangement  into  the  same  field  with 
the  star  whose  brightness  is  to  be  measured. 

Photometric  observations  in  many  cases  require  large  and  some- 
what uncertain  corrections,  especially  for  the  absorption  of  light  by 
the  atmosphere  at  different  altitudes,  and  the  final  results  of  different 
observers  naturally  fail  of  absolute  accordance.  Still  the  agreement 
between  the  different  recent  catalogues  is  remarkably  close,  generally 
within  one  or  two  tenths  of  a  magnitude. 

828.  The  Meridian  Photometer.  —  This   instrument,   contrived   and 
used  by  Professor  Pickering  in  the  observations  of  the  Harvard  Photometry, 
consists  of  a  telescope  with  two  object-glasses  side  by  side.     The  telescope 
is  pointed  nearly  east  and  west,  and  in  front  of  each  object-glass  is  placed 
a  silvered  glass  mirror  at  an  angle  of  45°.     One  of  the  mirrors  is  so  set  as 


PHOTOMETRY   AND   PHOTOGRAPHY.  513 

to  bring  the  rays  of  the  pole  star  to  one  object-glass ;  the  other  mirror  is 
capable  of  being  turned  around  the  optical  axis  of  the  telescope,  in  such  a 
way  as  to  command  a  star  at  any  part  of  the  meridian,  and  bring  its  light 
into  the  other  object-glass.  At  the  eye  end  is  placed  a  double  refraction 
polarization  apparatus,  which  gives  an  image  of  the  pole  star  polarized  in 
one  plane,  and  an  image  of  the  star  to  be  observed  polarized  in  a  plane  at 
right  angles,  both  visible  at  the  same  time  in  the  same  eye-piece.  In  front 
of  the  eye-piece  is  a  Nicol  prism.  On  looking  into  the  instrument  the 
observer  sees  two  stars,  the  pole  star  at  rest,  the  other  moving  along  as  in 
a  transit  instrument.  He  simply  turns  the  Nicol  until  the  images  are 
equalized,  setting  the  Nicol  at  all  the  four  different  positions  which  will 
produce  the  effect,  and  reading  the  graduated  circle  C.  The  whole  opera- 
tion consumes  not  more  than  a  minute,  with  the  help  of  an  assistant  to 
record  the  numbers  as  read  off.  The  "Harvard  Photometry"  (usually 
referred  to  simply  as  "H.  P.,")  was  made  by  means  of  an  instrument  with 
object-glasses  only  two  inches  and  a  half  in  diameter.  An  instrument  with 
four-inch  lenses  was  used  at  Cambridge  in  measuring  the  magnitudes  of 
all  the  nearly  80000  stars  of  Argelander's  Durchmusterung,  which  are  of 
the  eighth  magnitude  or  brighter,  and  has  been  sent  to  Peru,  where  it  is 
to  complete  the  photometry  of  the  southern  heavens. 

829,  Photometry  by  means  of  Photography. — It  has  been  found  that, 
excepting  a  few  strongly  colored  stars,  the  intensity,  or  more  simply  the  size, 
of  the  image  of  a  star  formed  upon  a  photographic  plate  may  be  used  as  a 
measure  of  its  brightness  as  compared  with  other  stars  taken  on  the  same 
plate,  or  on  similar  plates  similarly  exposed.  The  comparison  becomes 
easier  and  more  accurate  if  the  photographic  telescope  is  not  made  to 
follow  the  stars  exactly,  but  is  allowed  to  lag  a  little,  so  that  the  star  forms 
a  "  trail."  It  will,  therefore,  be  possible  to  use  the  plates  of  the  great  pho- 
tographic star  campaign  to  determine  star  magnitudes  as  well  as  positions. 
But  these  magnitudes  are  " photographic,"  not  visual;  certain  stars,  for 
instance,  that  are  hardly  visible  to  the  naked  eye,  photograph  as  bright 
stars,  and  there  are  others  —  red  stars  —  that  are  abnormally  faint  on  the 
plate.  The  exceptions  are  numerous  enough  to  make  it  necessary  to  use 
photographic  magnitudes  with  caution.  In  the  study  of  variable  stars, 
however,  the  method  may  be  used  with  some  great  advantages. 

.  830.  Star  Colors  and  their  Effects  on  Photometry. — The  stars 
differ  considerably  in  color.  The  majority  are  of  a  very  pure  white, 
like  Sirius  and  the  sun,  but  there  are  not  a  few  of  a  yellowish  hue, 
like  Capella,  or  reddish,  like  Arcturus  and  Antares  ;  and  there  are 
some,  mostly  small  stars,  which  are  as  red  as  garnets  and  rubies. 
We  also  have,  associated  with  larger  ones  in  double-star  systems, 


,514  SPECTRUM   PHOTOMETRY. 

numerous  small  stars  which  are  strongly  green  or  blue  ;  and  there 
are  a  few  large  isolated  stars,  which,  like  Vega,  are  of  a  decidedly 
bluish  tinge. 

These  differences  of  color  embarrass  photometric  measurements  made  by 
either  of  the  methods  described,  because  it  is  impossible  to  make  a  red  star 
look  identical  with  a  blue  one  by  any  mere  increase  or  diminution  of  bright- 
ness, and  because  different  observers  will  differ  in  setting  the  wedge  of  an 
extinction  photometer  according  to  the  color  of  the  star.  Some  eyes  are 
abnormally  sensitive  to  blue  light,  some  to  red.  To  the  writer,  for  instance, 
Vega  is  decidedly  superior  to  Arcturus,  while  the  majority  of  observers  see 
the  difference  as  decidedly  the  other  way. 

831.  Spectrum  Photometry.  —  The  only  completely  satisfactory  and 
scientific  method  would  be  to  compare  the  spectra  of  the  stars  with  some 
standard  spectrum,  say  that  of  the  pole  star,  dividing  the  spectrum  into  a  con- 
siderable number  of  portions,  and  determining  and  recording  the  amount 
of  light  in  each  portion  of  the  spectrum  as  compared  with  homologous 
parts  of  the  standard  spectrum.     This,  of  course,  would  immensely  increase 
the  work  of  comparing  the  brightness  of  the  stars  ;  but  it  is  quite  feasible 
to  do  it  for  a  few  hundred  of  the  brighter  ones,  and  it  would  be  well  worth 
accomplishment.     If  we  ever  succeed  in  getting  photographic  plates  equally 
sensitive  to  rays  of  all  wave  length,  photography  would  answer  the  purpose 
well. 

The  day  may  come  when  we  shall  have  "bolometers"  or  "radiometers" 
sufficiently  delicate  to  enable  us  to  extend  our  measures  through  the  whole 
extent  of  the  spectrum  —  the  invisible  portions  as  well  as  the  visible  —  and 
so  determine  the  total  "energy  "  sent  to  us  from  a  star:  but  we  are  far  enough 
from  it  at  present. 

It  is  worth  noting,  also,  that  Minchin  has  recently  met  with  encouraging 
success  in  attempting  to  measure*  the  luminosity  of  stars  by  its  effect  in 
changing  the  electrical  resistance  of  selenium,  and  the  method  may  ulti- 
mately develop  into  something  valuable. 

832.  Starlight  compared  with  Sunlight.  —  The  light  received  from 
Vega  is  about  ^o  (TirWuuoTr  (one  f°rty  thousand  millionth)  of  that 
from  the  sun,  according  to  the  determinations  of  Zollner  and  others. 
The  measurement  is  not  easy,  and  must  be  taken  as  having  a  very 
considerable  margin  of  error. 

Sirius  is  nearly  equivalent  to  six  of  Vega,  its  light  being  about 


The  light  of  the  standard  first  magnitude  star  may  be  taken  as 
about  half  that  of  Vega,  or  ^^^i^^^.  of  sunlight  ;  so  that  on 
the  absolute  scale  the  sun  is  reckoned  as  of  the  —  26.3  "magnitude." 


TOTAL   LIGHT    OF    THE    STARS.  515 

Since  the  light  of  a  sixth-magnitude  star  is  only  ^^  of  that  of  a 
standard  first-magnitude,  it  follows  that  it  would  require  8  000000- 
000000  stars  of  the  6th  magnitude  to  give  us  sunlight. 


833,  Total  Light  of  the  Stars.  —  Assuming   what   is   roughly, 
though  not  exactly,  true,  that  Argelander's  magnitudes  follow  the 
standard  scale,  it  appears  that  the  324000  stars  north  of  the  equator 
enumerated  in  his  Durchmusterung  give  a  light  about  equal  to  that 
of  240  first-magnitude  stars;  but  it  is  noticeable  that  the  aggregate 
amount  of  light  given  by  the  stars  in  each  of  the  fainter  magnitudes 
increases  rapidly. 

The  following  is  the  estimate,  substantially  according  to  Newcomb : 

10  stars  (above  the  2d  magnitude)  =      6.0  first-magnitude  stars. 

37  «  from  2d    to  3d  «  =     7.3  «  « 

122  «  "      3d    to  4th  «  =     9.6  «  « 

310  "  «      4th  to  5th  "  =     9.8  «  « 

1016  "  "      5th  to  6th  "  =   12.7  «  « 

4322  "  "      6th  to  7th  «  =21.6  «  « 

13593  «  «      7th  to  8th  «  =  27.1  «  « 

57960  "  "      8th  to  9th  «  =  46.  «  « 

247544  «  «      9th  to  9£  «  =100.  ±  «  « 

Total =240. 

How  much  to  add  for  the  still  smaller  magnitudes  is  very  uncertain. 
Beyond  the  tenth  magnitude  the  number  of  small  stars  does  not  increase 
proportionately  fast,  so  that  if  we  could  carry  on  the  account  of  stars  to  the 
twentieth  magnitude,  it  is  practically  certain  that  we  should  not  find  the 
total  light  of  the  aggregate  stars  of  each  succeeding  magnitude  increasing 
at  any  such  rate  as  from  the  seventh  to  the  tenth.  Perhaps  it  would  be  a 
not  unreasonable  estimate  to  put  the  total  starlight  of  the  northern  hemi- 
sphere as  equivalent  to  about  1500  first-magnitude  stars,  or  that  of  the 
whole  sphere  at  3000.  This  would  make  the  total  starlight  on  a  clear 
night  about  ^  of  the  light  of  the  full  moon,  and  about  ^^^1^^  that  of 
the  sun.  The  light  from  the  stars  which  are  visible  to  the  naked  eye 
would  not  be  as  much  as  ^V  °f  tne  whole.  Newcomb  (1902)  estimates  the 
total  light  of  all  the  stars  as  about  equal  to  600  stars  of  magnitude  zero 
—  about  one-fourth  of  the  estimate  above  made. 

834.  Heat  from  the  Stars.  —  Probably  the  heat  of  a  star  bears  to 
solar  heat  about  the  same  ratio  as  that  of  their  lights ;  it  is  at  the 
very   limit   of  perception.      Stone   (1869)    thought   his   thermopile 


516  AMOUNT   OF   LIGHT   FROM   CERTAIN   STARS. 

showed  sensible  effects  of  heat  from  Arcturus  and  Vega;  but  later 
work  discredited  his  results.  About  1890  Boys  with  his  much  more 
sensitive  "  radio-micrometer  "  failed  in  obtaining  any  distinct  indica- 
tions. The  first  certain  success  was  reached  in  1898-1900  by  Prof. 
E.  F.  Nichols  at  the  Yerkes  Observatory  with  an  apparatus  twenty 
times  as  sensitive  as  that  of  Boys.  He  obtained  measurable 
"  deflections  "  from  Vega,  Arcturus,  Jupiter,  and  Saturn. 

Arcturus  appeared  to  give  2.2  times  as  much  heat  as  Vega;  Jupiter  4.7  as 
much;  and  Saturn  f  ;  Vega  about  as  much  as  a  standard  candle  9  miles  distant. 

835.  Amount  of  Light  emitted  by  Certain  Stars.  —  This,  of  course, 
is  something  vastly  different  from  that  received  by  us.  A  star  may 
emit  hundreds  of  times  as  much  light  as  the  sun,  and  yet,  if  the  star 
is  remote  enough,  the  amount  that  reaches  the  earth  will  be  only  an 
excessively  minute  fraction  of  sunlight.  If  I  be  the  amount  of  light 
that  we  receive  from  a  star,  expressed  in  terms  of  sunlight  at  the 
earth,  then  the  total  amount  of  light  emitted,  L,  is  given  by  the 
simple  equation,  f 

L  =  I  X  D2,  or  (nearly)  4000  000000  X  I  X  Dy2. 

D  being  the  distance  of  the  star  in  astronomical  units,  and  Dy,  in  "light- 
years,"  while  L  is  expressed  in  terms  of  the  sun's  light  emission. 

Turning  to  the  table  of  stellar  parallax  (Appendix,  Table  IV.), 
we  find  that,  according  to  Gill  &  Elkin,  D  for  Sirius  equals  529000; 

5290002 
whence,  for  Sinus,     L  =  70(J()  OQQOOQ  ==  40.0  ; 

that  is,  the  light  emitted  by  Sirius  is  forty  times  as  much  as  that- 
emitted  by  the  sun. 

Similarly  for  the  pole-star  (p  =  0".07),  L  =  68  ;  for  Vega  (p  =  O'MG),  L  — 
44  ;  a  Centauri  (p  =  0".7o),  L  =  1.9  ;  70  Ophiuchi  (p  =  0".25),  L  =  0.41  ; 
61  Cygni  (p  =  0".40),  L  =  &  ;  21258  LI.  (p  =  0".26),  L  =  y^-1 

The  companion  of  Sirius  is  a  little  star  of  the  ninth  magnitude, 
which  forms  a  double-star  system  with  Sirius  itself.  The  light 
emitted  by  this  companion  does  not  exceed  T^n  o  ^na^  °f  Sirius. 


836.     Causes  of  Differences  of  Brightness  in  Stars.  —  It  used  to  be 
thought  that  the  stars  were  all  very  much  alike  in  magnitude  and 

1  In  making  this  calculation  the  magnitudes  of  the  Harvard  Photometry  were 
used. 


REAL    DIAMETER    OF    STARS.  517 

constitution;  not,  indeed,  without  considerable  differences,  but  as 
much  resembling  each  other  as  do  individuals  of  the  same  race.  It 
is  now  quite  certain  that  this  is  not  the  case,  as  is  obvious  from  the 
short  list  of  actual  light  emissions  just  given. 

If  the  stars  were  all  alike,  all  the  differences  of  apparent  bright- 
ness would  be  traceable  simply  to  differences  of  distance;  but  as  the 
facts  are,  we  have  to  admit  other  causes  to  be  equally  effective. 
The  differences  of  brightness  are  due,  first,  to  difference  of  distance; 
second,  to  difference  of  dimensions,  or  of  light-giving  area ;  third,  to 
difference  in  the  brilliance  of  the  light-giving  surface,  depending  upon 
difference  of  temperature  and  constitution.  There  are  stars  near  and 
remote,  large  and  small,  intensely  incandescent  and  barely  glowing 
with  incipient  or  failing  light. 

As  Bessel  puts  it,  there  is  no  reason  why  there  may  not  be  "as 
many  dark  stars  as  bright  ones,"  and,  as  we  shall  soon  see,  there  is 
now  positive  evidence  that  they  are  really  numerous.  The  com- 
panion of  Sirius,  though  only  giving  about  -TZVOV  Part  as  much 
light  as  Sirius  itself,  is  nearly  i  as  heavy ;  so  that,  mass  for  mass, 
it  cannot  be  ^-^  part  as  luminous. 

When  we  compare  stars  by  the  thousand,  we  can  say  of  the  tenth- 
magnitude  stars,  for  instance,  as  compared  with  the  fifth,  that  as  a 
class  they  are  more  remote;  and  also,  just  as  truly,  that  their  average 
diameters  are  smaller,  and  also  that  their  surfaces  are  less  brilliant; 
but  we  must  be  careful  not  to  make  any  assertions  of  this  sort 
regarding  any  one  star  of  the  tenth  magnitude  compared  with  a 
particular  individual  of  the  fifth,  unless  we  have  some  absolute 
knowledge  of  their  relative  distances.  The  faint  star  may  be  the 
larger  of  the  two,  or  the  hotter,  or  the  nearer.  We  must  know 
something  beyond  their  relative  "  magnitudes"  before  it  is  possible 
to  settle  such  questions. 


837.  Real  Diameter  of  Stars. — As  to  the  apparent  angular 
diameter,  we  can  only  say  negatively  that  it  is  insensible,  in  no  case 
being  known  to  reach  0".01.  If  there  be  a  star  of  the  same  diam- 
eter as  our  sun,  at  such  a  distance  that  its  parallax  equals  one 

1924" 

second,  its  apparent  diameter  must  be  •     [The  sun's  mean 


angular  diameter  is  1924"  (Art.  276).]  This  equals  0".0093  —  a 
quantity  far  too  small  to  be  reached  by  any  direct  measurement, 
especially  since,  even  in  the  Lick  telescope,  the  "  spurious  "  disc  of 


518  VAEIABLE   STARS. 

a  star  has  a  diameter  of  nearly  |",  and  in  smaller  telescopes  is  much 
larger  (about  0".4  in  a  ten-inch  telescope). 

There  is  a  theoretical  connection  between  the  diameter  of  the  diffraction 
rings  seen  around  the  image  of  a  star  in  the  telescope,  and  the  real  (as 
opposed  to  the  spurious)  diameter  of  the  image;  by  comparing,  therefore,  the 
actual  size  of  the  rings  with  the  size  they  should  have  if  the  star  were  an 
absolute  optical  point,  we  might  hope  to  get  a  determination  of  the  star's 
angular  diameter.  Thus  far,  however,  no  satisfactory  results  have  been 
obtained.  Michelson  has  proposed  a  somewhat  similar  method  based  on  the 
diffraction  fringes  produced  when  a  pair  of  parallel  slits  are  placed  in  front 
of  the  object-glass  of  a  telescope.  But  the  calculation  of  the  diameter  of  the 
stellar  disc  depends  upon  the  assumption  that  it  is  uniformly  luminous  all  over, 
or,  if  not,  that  we  know  the  law  of  distribution, —  assumptions  by  no  means 
safe. 

In  a  single  case,  that  of  the  variable  star  Algol  (Art.  848),  the  diameter 
has  actually  been  determined  by  the  peculiarities  of  its  variation  combined 
with  spectroscopic  observations  (Art.  851);  and  quite  possibly  other  similar 
cases  may  be  found  before  very  long.  Algol  has  a  diameter  of  about 
1 060000  miles. 

VARIABLE  STABS. 

838.  A  close  examination  shows  that  many  stars  change  their 
brightness,  and  are  called  " variable"     The  variable  stars  may  be 
classified *  as  follows :  — 

I.    Cases  of  slow  continuous  change. 

II.  Irregular  fluctuations  of  light :  alternately  brightening  and 
darkening  without  any  apparent  law. 

III.  Temporary  stars,  which  blaze  out  suddenly  and  then  dis- 
appear. 

IV.   Periodic  stars  of  long  period  (two  months  to  two  years). 
V.   Periodic  stars  of  short  period  (a  few  hours  to  three  weeks). 

VI.  Periodic  stars  in  which  the  variation  is  like  that  which 
would  be  the  result  of  an  eclipse  by  some  intervening 
body  —  the  Algol  type. 

839,  I.    GRADUAL  CHANGES.     On  the  whole,  the  changes  in  the 
brightness  of  the  stars  since  the  time  of  Hipparchus  and  Ptolemy 
have  been  surprisingly  small.     There  has  been  no  general  increase 

1  This  classification  is  substantially  that  of  Professor  Pickering,  slightly 
modified,  however,  by  Houzeau. 


MISSING   STARS.  519 

or  decrease  in  the  brightness  of  the  stars  as  a  whole;  and  there  are 
few  cases  where  any  individual  star  has  altered  its  brightness  by  a 
half  or  even  a  quarter  of  a  magnitude.  The  general  appearance  of 
the  sky  is  the  same  as  it  was  2000  years  ago;  so  that  notwithstand- 
ing all  the  effect  of  proper  motions  in  the  meantime  and  the  whole 
amount  of  the  variation  that  has  taken  place  in  the  brightness  of 
the  stars,  there  is  no  doubt  that  if  either  of  these  old  astronomers 
were  to  come  to  life  he  would  immediately  recognize  the  familiar 
constellations. 

There  are  a  few  instances,  however,  in  which  it  is  almost  certain 
that  change  has  taken  place  and  is  going  on.  In  the  time  of  Eratos- 
thenes the  star  in  the  "  claw  of  the  Scorpion  "  (now  ft  Librae)  was 
reckoned  the  brightest  in  the  constellation.  At  present,  it  is  a  whole 
magnitude  below  Antares,  which  is  now  much  superior  to  any  star 
in  the  vicinity.  So  when  the  two  stars  Castor  and  Pollux  in  the 
constellation  Gemini  were  lettered  by  Bayer,  the  former,  a,  was 
brighter  than  Pollux,  which  was  lettered  /?;  but  fi  is  now  notably 
the  brighter  of  the  two.  Taking  the  whole  heavens,  we  find  a 
considerable  number  of  such  cases;  perhaps  a  dozen  or  more. 

840.  Missing  Stars.  —  It  is  a  common  belief  that  since  accurate 
star-catalogues  began  to  be  made,  many  stars  have  disappeared  and 
not  a  few  new  ones  have  come  into  existence.     While  it  would  not 
do  to  deny  absolutely  that  anything  of  the  kind  has  ever  happened, 
it  is  certainly  unsafe  to  assert  that  it  has. 

There  are  a  considerable  number  of  cases  where  stars  are  now  missing 
from  the  older  catalogues  as  published,  —  nearly,  if  not  quite,  a  hundred,  — 
but  in  almost  every  instance  examination  of  the  original  observations  shows 
that  the  place  as  printed  was  a  mistake  of  some  sort  which  can  now  be 
traced,  —  sometimes  a  mistake  of  the  recorder,  sometimes  in  the  reduction 
of  the  observation,  and  sometimes  of  the  press.  In  a  few  cases  the  star 
observed  was  a  planet  (Uranus,  Neptune,  or  an  asteroid) ;  and  in  some  cases 
the  missing  star  may  have  been  a  "temporary  star,"  as,  for  instance,  55 
Herculis,  which  was  observed  by  the  elder  Herschel.  So  many  of  the  missing 
stars  are  now  satisfactorily  explained  that  it  is  natural  to  suppose  that  the 
few  cases  remaining  are  of  the  same  sort. 

There  is  no  known  instance  of  a  new  star  appearing  and  remaining 
permanently  bright. 

841.  II.    STARS   THAT   EXHIBIT  IRREGULAR   FLUCTUATIONS   IN 
BRIGHTNESS.     The  most  conspicuous  example  of  this  class  is  the 


520 


TEMPORARY    STARS. 


strange  star  y  Argus  (not  visible  in  the  United  States).  This  star 
ranges  all  the  way  from  the  zero  magnitude  (in  1843,  when,  accord- 
ing to  Sir  John  Herschel,  it  was  brighter  than  every  star  but 
Sirius)  down  to  the  seventh  magnitude,  which  is  its  present  bright- 
ness and  has  been  ever  since  1865.  It  is  often  called  -^  Carinm,  the 
constellation  Argo-Navis  being  subdivided  into  Puppis,  Vela,  and 
Carina. 

Between  1677  (when  it  was  observed  by  Halley  as  of  the  fourth  magni- 
tude) and  1800,  it  oscillated  in  brightness,  so  far  as  we  can  judge  from  the 
few  observations  extant,  between  the  fourth  and  second  magnitudes.  About 
1810,  it  rose  rapidly  in  brightness,  and  between  1826  and  1850  it  was  never 
below  the  standard  first  magnitude.  When  brightest,  in  1843,  it  was  giving 
more  than  25000  times  as  much  light  as  in  1865.  A  singular  fact  is  that 
the  star  is  in  the  midst  of  a  nebula  which  apparently  sympathizes  with  it  to 
some  extent  in  its  fluctuations.  (There  are  other  instances  of  connection 
of  nebulae  with  variable  or  temporary  stars,  as  will  appear  later  on.) 
Fig.  220  represents  the  "light-curve"  of  this  object  from  1800  to  1870, 
according  to  Loomis.  It  is  barely  possible,  though  hardly  probable,  that 
the  star  may  be  periodic  with  a  period  of  about  70  years ;  but  if  so,  it  is 
unique. 


1810       1830 


1830       18.40 


1850 


1870 


FIG.  220.  —  Light-Curve  of  ij  Argus  according  to  Loomis. 

a  Orionis,  a  Herculis,  and  a  Cassiopeise  behave  in  a  somewhat 
similar  manner,  only  the  whole  range  of  variation  in  their  brightness 
is  less  than  a  single  magnitude,  and  the  oscillations  never  extend 
over  more  than  two  or  three  years.  The  catalogue  of  variable  stars 
shows  a  considerable  number  of  other  similar  cases. 


842.     III.    TEMPORARY  STARS.     There  are  fifteen  well  authen- 
ticated cases  in  which  new  stars  have  appeared  temporarily,  —  that 


TEMPORARY    STARS.  521 

is,   for  a  few  weeks  or  months,  —  blazing  up  suddenly  and  then 
gradually  fading  away.     The  list1  is  as  follows,  up  to  1898:  — 

1.  134  B.C.     The  star  of  Hipparchus. 

2.  389  A.D.     A  star  in  Aquila. 

3.  1572  A.D.  Tycho's  star  in  Cassiopeia. 

4.  1600  A.D.  P  Cygni,  3d  magnitude,  observed  by  Jansen. 

5.  1604  A.D.  Kepler's  star  in  Ophiuchus. 

6.  1670  A.D.  11  Vulpeculae,  3d  magnitude,  observed  by  Anthelm. 

7.  1848  A.D.  A  star  of  the  5th  magnitude,  observed  by  Hind  —  also  in 

Ophiuchus. 

8.  1860  A.D.     T  Scorpii,  7th   magnitude,  in   the  star  cluster  M  80, 

observed  by  Auwers. 

9.  1866  A.D.     T  Coronse-Borealis,  2d  magnitude. 

10.  1876  A.D.     Nova  Cygni,  3d  magnitude. 

11.  1885  A.D.     A  star  in  the  nebula  of  Andromeda,  6th  magnitude. 

12.  1892-93  A.D.     Nova  Aurigse,  4th  magnitude. 

13.  1893  A.D.     Nova  Normae,  7th  magnitude. 

14.  1895  A.D.     Nova  Carinae  (Argus),  8th  magnitude. 

15.  1898  A.D.     Nova  Sagittarii,  5th  magnitude. 

As  regards  the  first  of  these  stars,  we  know  almost  nothing.  Hipparchus 
has  left  no  record  of  its  position  or  brightness ;  but  the  Chinese  annals 
mention  a  star  as  appearing  in  Scorpio  at  just  that  date,  and  probably  the 
same  object;  though  the  Chinese  observations  may  refer  to  a  comet.  The 
appearance  of  this  new  star  led  Hipparchus  to  form  his  catalogue  of  stars. 

As  to  the  second,  possibly  a  comet,  we  know  hardly  more. 

843.  The  third  is  justly  famous.  When  it  was  first  seen  by 
Tycho  in  November,  1572,  it  was  already  brighter  than  Jupiter, 
having  probably  appeared  a  few  days  previously.  It  very  soon 
became  as  bright  as  Venus  herself,  being  even  visible  by  day.  Within 
a  week  or  two  it  began  to  fail,  but  continued  visible  to  the  naked 
eye  for  fully  sixteen  months  before  it  finally  disappeared.  It  is  not 
certain  whether  it  still  exists  or  not  as  a  telescopic  star:  Tycho 
determined  its  position  with  as  much  accuracy  as  was  possible  to 
his  instruments;  and  there  are  a  number  of  small  stars,  any  one  of 
which  is  so  near  to  Tycho's  place  that  it  might  be  the  real  object. 

There  has  been  an  entirely  unfounded  notion  that  this  star  may  have 
been  identical  with  the  "  Star  of  Bethlehem,"  it  being  imagined  that  the 
star  is  periodically  variable,  with  a  period  of  314  years.  If  so,  it  might  have 
been  expected  to  reappear  in  1886,  and  it  was  so  expected  by  certain  persons 

1  See  Addendum  B,  following  page  580. 


522  TEMPORARY   STARS. 

"  as  a  sign  of  the  second  coming  of  the  Lord."  It  is  difficult  to  see  how  the 
idea  came  to  be  so  generally  prevalent  as  it  certainly  has  been.  Probably 
every  astronomer  of  any  note  has  received  hundreds  of  letters  on  the  subject. 
At  Greenwich  a  printed  circular  was  prepared  and  sent  out  as  a  reply  to  such 
inquiries. 

The  fifth,  star,  observed  by  Kepler,  was  nearly,  though  not  quite, 
as  bright  as  that  of  Tycho,  and  lasted  longer  —  fully  two  years.  It 
also  has  disappeared  so  that  it  cannot  now  be  identified. 

844.  The   ninth   star   excited    much    interest.      It   blazed   out 
between   the   10th   and   12th   of   May   as   a    star   of  the   second 
magnitude,  remained  at  its  maximum  for  three  or  four  days,  and 
then,  in  five  or  six  weeks,  faded  away  to  its  original  faintness,  for 
it  now  is,  and  was  before  the  outburst,  a  nine  and  one-half  magnitude 
star  on  Argelander's  catalogue,  with  nothing  noticeable  to  distinguish 
it  from  its  neighbors.     While  at  the  maximum  its  spectrum  was 
carefully  studied  by  Huggins,  and  exhibited  brightly  and  strongly 
the  bright  lines  of  hydrogen,  just  as  if  it  were  a  sun  like  our  own, 
but  entirely  covered  with  outbursting  "prominences"  of  incandescent 
hydrogen. 

The  tenth  star  had  a  very  similar  history.  It  also  rose  to  its  full 
brightness  (second  magnitude)  on  November  24,  within  four  hours 
according  to  Schmidt,  remained  at  a  maximum  for  only  a  day  or 
two,  and  faded  away  to  invisibility  within  a  month.  But  it  still 
exists  as  a  very  minute  telescopic  star  of  the  fifteenth  magnitude. 
It  was  also  spectroscopically  studied  by  several  observers  (by  Vogel 
especially)  with  the  strange  result  that  the  spectrum,  which  at  the 
maximum  was  nearly  continuous,  though  marked  by  the  bright  lines 
of  hydrogen  and  by  bands  of  other  unknown  substances,  lost  more 
and  more  of  this  continuous  character,  until  at  last  it  became  a 
simple  spectrum  of  three  bright  lines  like  that  of  a  nebula. 

845.  The  eleventh  of  these  temporary  stars  was  very  peculiar 
in  one  respect;  not  in  its  brightness,  for  it  never  exceeded  the  six 
and  one-half  magnitude,  but  because  it  appeared  right  in  the  midst 
of  the  great  nebula  of  Andromeda,  only  12"  or  13"  distant  from  the 
nucleus.     It  came  out  suddenly  like  all  the  others,  and  faded  away 
gradually  in  about  six  months,  to  absolute  extinction  so  far  as  any 
existing  telescope  can  show.    It  showed  under  photometric  measure- 
ments many  fluctuations  in  brightness,  not  losing  its  light  smoothly 


TEMPORARY   STARS. 


523 


and  regularly,  but  in  a  rather  paroxysmal  manner.  Its  spectrum, 
even  when  brightest,  was  simply  continuous  without  anything  more 
than  the  merest  trace  of  bright  lines  in  it.  The  eighth  star  on  the 
list  resembled  it  in  the  fact  that  it  appeared  in  the  midst  of  a  star 
cluster. 

845*.  In  Jan.  1892  a  twelfth  "  Nova  "  appeared  in  the  constella- 
tion of  Auriga,  which  at  its  brightest,  about  Feb.  5th,  was  a  star  of 
the  4-J-  magnitude.  Its  spectrum  (Fig.  220*)  was  very  peculiar,  show- 
ing a  great  number  of  bright  lines,  especially  those  of  hydrogen  and 
with  them  also  the  dark  lines  of  the  same  substances.  The  bright 
and  dark  lines  were  displaced  relatively  to  each  other  as  if  they 
were  respectively  due  to  at  least  two  different  masses  of  gas,  in 
swift  relative  motion 1  at  the  rate  of  something  like  500  miles  a 
second,  —  the  "  bright-line  "  mass  receding  from  us,  and  the  other 
approaching. 

In  the  autumn  the  star,  which  had  sunk  to  the  llth  magnitude,  again 
brightened  up  to  about  the  9th,  and  then  the  spectrum  was  found  to  be 
almost,  if  not  absolutely,  identical  with  that  of  a  planetary  nebula,  but 
finally,  according  to  Campbell,  in  1903  became  simply  continuous. 


FIG.  220  *  is  from  a  photograph  by  Frost  at  Potsdam. 

The  "Novae"  of  1883,  '95  and  '98  are  peculiar  in  the  fact  that  they 
were  detected  by  photography,  having  been  recognized  by  Mrs. 
Fleming  of  the  Harvard  College  Observatory  upon  both  the  chart 
plates  and  spectrum  photographs  taken  at  the  Harvard  Station  in 


1  But  see  Art.  802*.  It  is  not  certain  that  the  displacements  of  the  lines  may 
not  have  been  due,  partly  at  least,  to  pressure-effects  accompanying  explosions 
or  eruptions  or  possibly  to  '•'•anomalous  refraction."  See  Addendum  B. 


524 


LONG-PEKIOD    VARIABLES. 


South  America.  Their  spectra  very  precisely  resembled  that  of 
Nova  Aurigye,  showing  the  same  duplex  combination  of  bright  lines 
with  dark.  It  now  seems  very  probable  that  the  "  new  stars " 
would  be  not  very  uncommon  if  the  small  stars  could  all  be  closely 
watched;  and  it  is  clear  that  there  are  important  physical  resem- 
blances between  them;  but  the  phenomena  are  not  yet  clearly 
explained. 

846.  IV.  LONG-PERIOD  VARIABLES,  o  CETI  TYPE.  These  objects 
resemble  the  temporary  stars  in  rapidly  brightening  up  for  a  short 
time  and  then  fading  back  to  the  original  condition;  but  they  do  it 
periodically.  The  periods  are  generally  of  considerable  length,  from 
six  months  to  two  years;  but  they  are  very  apt  to  be  considerably 
irregular,  not  unfrequently  to  the  extent  of  several  weeks. 


»: 


«J.5 


100 


200 


300 


400 


500 


600  days 


"-^wmmiM^p  mdm-^m. 


3.5 


M¥& 


- 


10 


15 


20  25  30  days 


;  5     ?  «  >  fi     4 


6d. 


FIG.  221.  —Light-Curves  of  Variable  Stars. 


The  star  o  Ceti  (often  called  Mira,  that  is,  "the  Wonderful")  may 
be  taken  as  the  type  of  this  class.  Its  variability  was  discovered  by 
Fabricius  in  1596.  During  most  of  the  time  it  remains  invisible  to 
the  naked  eye,  falling  below  the  9th  magnitude  at  its  minimum,  but 
once  in  about  eleven  months  it  runs  up  to  the  fourth,  third,  or  even 


SHORT-PERIOD    VARIABLES.  525 

the  second  magnitude,  and  then  back  again.  Its  brightness  increases 
more  rapidly  than  it  fails,  and  it  remains  at  its  maximum  for  a  week 
or  ten  days,  and  sometimes  much  longer.  The  maxima  vary  very 
much  in  brightness,  and  are  frequently  several  weeks  ahead  of,  or 
behind,  the  computed  time.  At  the  maximum  its  spectrum  is  very 
beautiful,  containing  a  large  number  of  intensely  bright  lines,  most 
of  which  are  due  to  hydrogen,  though  some  of  them  are  still 
unidentified.  Its  light-curve  is  A,  in  Fig.  221.  A  large  majority 
of  the  known  variables  belong  to  this  class.  Nearly  all  of  them 
are  notably  reddish  in  color,  and  most  of  them  show  a  colonnaded 
spectrum  marked  with  bright  lines. 

847.  V.     SHORT-PERIOD  VARIABLES.1    TYPE  OF  rj  AQUIL^E  AND 
(3  LYR^E.     In  these  the  periods  range  from  three  and  one-fourth 
hours  (that  of  XX  Cygni,  the  shortest  known  at  present)  to  three 
weeks,  and  the  light  of  the  star  fluctuates  all  the  time.     In  many 
cases  there  are  two  or  more  minima  in  a  complete  period,  accom- 
panied by  complicated  spectroscopic  phenomena  of  bright  and  dark 
lines,  which  shift  their  places,  and  double  and  undouble  themselves 
in  a  very  interesting  and  significant  way.     (See  Art.  872*.)     The 
light-curves  of  rj  Aquilse  and  ft  Lyrse  are  given  at  B,  Fig.  221. 

848.  VI.    VARIABLES  OF  THE  ALGOL  TYPE.*    The  sixth  and  last 
class  consists  of  stars  which  seem  to  suffer  a  partial  eclipse  at  short 
intervals.     Of  this  type  of  stars,  Algol,  or  ft  Persei,  may  be  taken 
as  the  most  conspicuous  representative.    Its  period  is  2d  20h  48m  55s. 4, 
which  is  subject  to  almost  no  variation,  except  certain  slow  changes 
that  appear  to  be  the  result  of  some  unknown  disturbance.     During 
most  of  the  time  the  star  remains  of  the  second  magnitude.    At  the 
time  of  obscuration  it  loses  about  five-sixths  of  its  light,  falling  to 
the  fourth  magnitude  in  about  four  and  one-half  hours,  remaining 
at  the  minimum  for  about  twenty  minutes,  and  then  in  three  and 
one-half  hours  recovering  its  original  condition.     During  the  mini- 
mum the  spectrum  undergoes  no  considerable  change  in  its  appear- 
ance, but  during  the  whole  period,  as  will  be  presently  explained, 
the  lines  in  it  regularly  shift  their  positions.     (See  Art.  851.) 

The  periods  are  all  short,  ranging  from  four  hours  to  thirteen 
days  five  hours.  About  sixty  stars  are  known  at  present  to  belong 
to  this  class. 

1  See  Addendum  C. 


526 


EXPLANATION    OF   VARIABILITY. 


849.  Explanation  of  Variability. — Evidently  no  single  explana- 
tion will  hold  for  all  the  different  classes.     For  the  gradually  pro- 
gressive changes  no  explanation  need  be  looked  for;  on  the  contrary, 
it  is  surprising  that  such  changes  are  no  greater  than  they  are,  for 
the  stars  are  all  growing  older. 

As  for  the  irregular  changes,  no  sure  account  can  yet  be  given  of 
them.  Where  the  range  of  variation  is  small,  as  it  is  in  most  cases, 
one  thinks  of  spots  on  the  surface  like  those  of  our  own  sun  (but 
much  more  extensive  and  numerous),  and  running  through  a  period 
just  as  our  sun  spots  do.  Let  a  star  with  such  spots  upon  it  revolve 
on  its  own  axis,  as  of  course  it  must  do,  and  in  the  combination  we 
have  at  least  a  possible  explanation  of  a  great  proportion  of  all  the 
known  cases,  both  the  irregular  variables  and  the  regularly  periodic. 

Many  of  the  spectroscopic  phenomena  of  the  temporary  stars  and 
of  some  of  the  long-period  variables  closely  resemble,  on  a  slightly 
magnified  scale,  those  that  are  observed  in  the  solar  chromosphere 
and  prominences.  The  same  bright  lines  of  hydrogen  and  helium 
appear,  and  the  same  behavior  of  gaseous  masses,  distorting  and 
displacing  the  lines.  The  facts  strongly  suggest  in  these  cases  a 
theory  of  "  explosions  "  or  eruptions. 

850.  Collision  Theory.  —  For   the   temporary  stars,  and  those 

of  the  o  Ceti  type,  Sir 
Norman  Lockyer  (in  con- 
nection with  a  much  more 
extended  subject)  sug- 
gests a  "collision  theory," 
illustrated  by  Fig.  221*. 
The  fundamental  idea 
that  the  phenomena  of 
the  temporary  stars  may 
be  due  to  collisions  is  not 
new.  Newton  long  ago 
brought  it  out,  and  to  some 
extent  discussed  it;  but 
considering  the  probable 
diameters  of  the  stars  as 
compared  with  the  dis- 

FIG.  221* -The  Collision  Theory  of  Variable  Stars.       tanceg   between   them,  ft 

seems  impossible  that  collisions  could  have  been  frequent  enough  to 
account  for  the  number  of  temporary  stars  actually  observed. 


STELLAK   ECLIPSES.  527 

Lockyer,  however,  imagines  that  the  temporary  stars,  and  also 
variable  stars  of  the  o  Ceti  class,  are,  in  their  present  stage  of 
development,  not  compact  bodies,  but  only  pretty  dense  swarms  of 
meteorites  of  considerable  extent,  each  accompanied  by  another 
smaller  one  revolving  around  it  in  an  eccentric  orbit,  just  as  comets 
revolve  around  the  sun,  or  as  the  components  of  double  stars  revolve 
around  each  other.  He  supposes  that  the  perihelion  distance  is 
so  small  that  the  swarms  interpenetrate  and  pass  through  each 
other  at  the  perihelion,  which  could  happen  without  disturbing  the 
general  motion  of  either  of  the  two  meteoric  flocks ;  but  while  they 
are  thus  passing,  the  collisions  are  immensely  increased  in  number 
and  violence,  with  a  corresponding  increase  in  the  evolution  of 
light.  There  are  many  good  points  about  this  ingenious  theory, 
but  also  serious  objections  to  it  —  as,  for  instance,  the  great  irregu- 
larity of  the  periods  of  stars  of  this  class,  an  irregularity  which 
seems  hardly  consistent  with  such  an  orbital  revolution. 

851,  Stellar  Eclipses.  —  As  to  the  Algol  type,  the  natural  expla- 
nation is  by  means  of  an  eclipse  of  some  sort.  The  interposition  of 
a  more  or  less  opaque  object  between  the  observer  and  the  star,  —  a 
dark  companion  revolving  around  it,  —  would  produce  just  the  effect 
observed,  as  was  suggested  by  Goodricke  a  century  ago.  That  this 
is  really  the  case  has  now  been  practically  demonstrated  by  the 
spectroscopic  work  of  Yogel  in  1889,  who  found  by  the  method 
indicated  in  Art.  802  *  that  from  twelve  to  eighteen  hours  before 
the  obscuration,  Algol  is  receding  from  us  at  the  rate  of  nearly 
twenty-seven  miles  a  second,  while  after  the  minimum  it  approaches 
us  at  the  same  rate.  This  is  just  what  it  ought  to  do,  if  it  had  a 
large,  dark  companion,  and  the  two  were  revolving  around  their 
common  centre  of  gravity  in  an  orbit  nearly  edgewise  to  the  earth. 
When  the  dark  star  is  rushing  forward  to  interpose  itself  between 
us  and  Algol,  Algol  itself  must  be  moving  backwards,  and  vice  versa 
when  the  dark  star  is  receding  after  the  eclipse.  Vogel's  conclusions 
are,  that  the  distance  of  the  dark  star  from  Algol  is  about  3250000 
miles;  that  their  diameters  are  respectively  about  840000  and 
1 060000  miles  ;  that  their  united  mass  is  about  two-thirds  that  of 
the  sun ;  and  their  density  about  one-fifth  that  of  the  sun,  —  not 
much  greater  than  that  of  cork.  See  Fig.  222. 

Furthermore,  from  the  variations  in  the  observed  period  of  the 
star,  alluded  to  in  Art.  848,  combined  with  certain  minute  irregu- 


528 


NUMBER    AND    DESIGNATION    OF    VARIABLES. 


larities  in  its  '  proper  motion/  Chandler  has  shown  it  to  be  likely 
that  this  swiftly  moving  pair  is  itself  revolving  around  another 
distant  and  invisible  star  in  an  orbit  about  as  large  as  that  of 
Uranus,  with  a  period  of  about  130  years.  Tisserand,  however, 
suggests  a  different  explanation,  depending  on  a  slow  revolution  of 
the  apsides  of  the  orbit. 


FlG.  222.    System  of  Algol. 

In  the  case  also  of  8  Cephei  and  fi  Lyrae  (of  Class  V.)  the  observa- 
tions of  Belopolsky,  Lockyer,  and  others  have  made  it  nearly  certain 
that  two  or  more  bodies  in  orbital  revolution  are  concerned  in  their 
phenomena,  the  variations  in  the  light  being  in  part  at  least  due  to 
a  more  or  less  complete  eclipse,  but  also  in  part  to  other  (tidal  ?) 
interactions  which  are  not  yet  clear.  It  is  not  unlikely  that  all  the 
punctual  variables  (those  that  keep  accurate  time  in  their  light- 
changes)  may  turn  out  to  be  spectroscopic  binaries  (Art.  879). 

In  certain  cases  (Y  Cygni  and  Z  Herculis),  the  odd  and  even 
minima  occur  at  unequal  intervals,  indicating  a  very  eccentric  orbit. 

852.  Number  and  Designation  of  Variables.1  —  The  "Third  Cata- 
logue of  Variable  Stars/'  by  Dr.  S.  C.  Chandler,  published  in  1896, 
contains  393  stars  of  which  the  variation  is  regarded  as  certain. 

1  See  Addendum  C. 


RANGE    OF    VARIATION.  529 

About  300  are  unquestionably  periodic,  of  which  about  250  belong 
to  the  o  Ceti  type,  29  to  the  rj  Aquilse  class,  and  14  to  the  Algol 
group.  There  are  half  a  dozen  or  so  with  periods  apparently 
between  twenty-five  and  sixty  days,  leaving  it  doubtful  how  they 
should  be  classed.  About  thirty  are  distinctly  irregular  in  their  vari- 
ation, and  there  are  about  fifty  in  respect  to  which  the  periodicity 
is  not  yet  either  ascertained  or  disproved.  The  catalogue  is  followed 
by  a  second  list  of  154  stars  which  are  "  suspected "  of  variation. 
The  number  of  known  variables  had  nearly  doubled  since  Mr. 
Chandler  published  his  first  catalogue  in  1888,  and  it  is  still  growing 
rapidly.  The  new  variables  are  mostly  "  telescopic." 

Table  VI.  in  the  Appendix  presents  the  principal  data  for  the  naked-eye 
variables  which  are  visible  in  the  United  States. 

When  a  star  is  discovered  to  be  variable  which  previously  had  no 
special  appellation  of  its  own,  it  is  customary  to  designate  it  by  one 
of  the  last  letters  in  the  alphabet,  beginning  with  R.  Thus  R  Sagittarii 
is  the  first  discovered  variable  in  Sagittarius ;  S  Sagittarii  is  the  second ; 
T  Sagittarii,  the  third,  and  so  on.  But  we  have  P  and  Q  Cygni,  both 
"  temporaries." 

853.  Range  of  Variation.  —  In  many  cases  the  whole  range  is 
only  a  fraction  of  a  magnitude  (especially  among  the  more  newly 
discovered  variables),  but  in  a  great  number  it  extends  from  four 
to  eight  magnitudes,  the  maximum  brightness  exceeding  the  mini- 
mum by  from  fifty  to  a  thousand  times;  and  in  a  few  cases  the 
range  is    greater  yet.      Not  unfrequently  considerable  changes  of 
color  accompany  the  changes  of  brightness;  the  star  as  a  rule  being 
whiter  at  its  maximum,  and  frequently  showing  bright  lines  in  its 
spectrum. 

854.  Method  of  Observation.  —  There  is  no  better  way  than  that 
of  comparing  the  star  by  the  eye,  or  with  the  help  of  an  opera-glass 
or  small  telescope, with  surrounding  stars  of  about  the  same  bright- 
ness at  the  time  when  its  light  is  near  the  maximum  or  minimum ; 
noting  to  which  of  them  it  is  just  equal  at  the  moment,  and  also 
those  which  are  a  shade  brighter  or  fainter. 

It  is  possible  for  an  amateur  to  do  really  valuable  work  in  this  way,  by 
putting  himself  in  relation  with  some  observatory  which  is  interested  in  the 
subject.  The  observations  themselves  require  so  much  time  that  it  is  diffi- 
cult for  the  working  force  in  a  regular  observatory  to  attend  to  the  matter 
properly,  and  outside  assistance  is  heartily  welcomed  in  gathering  the  needed 


530  STAR    SPECTRA. 

facts.  The  observations  themselves  are  not  specially  difficult,  require  no 
very  great  labor  or  mathematical  skill  in  their  reduction,  and,  as  has  been 
said,  can  be  made  without  instruments;  but  they  require  patience,  assiduity, 
and  a  keen  eye. 

Photography  also  has  lately  come  to  the  front  as  a  most  effective 
method.  A  very  large  proportion  of  the  variables  discovered  within 
the  last  few  years  have  been  found  by  the  comparison  of  the  photo- 
graphic star-charts  made  at  Cambridge  and  at  the  Harvard  South 
American  Stations.  In  several  cases  the  photographed  spectrum  of 
a  star  has,  by.  its  peculiar  "  colonnaded  "  character  and  bright  lines, 
attracted  the  attention  and  marked  it  as  "  suspicious";  and  in  nearly 
every  case  the  suspicion  has  been  verified. 

854*.  Variable-Star  Clusters.1— One  of  the  most  interesting  and 
even  startling  results  of  stellar  photography  is  the  discovery  of 
variable-star  clusters,  announced  by  Professor  Pickering  in  1895. 
Attention  had  been  called  to  certain  clusters  by  the  visual  discovery 
of  one  or  two  variables  in  them,  and  Mr.  Bailey,  who  for  several 
years  has  been  in  charge  of  the  Southern  photographic  operations, 
soon  obtained  a  large  number  of  negatives  of  several  of  them,  and 
immediately  found  that,  while  many  clusters  show  no  variables, 
others  contain  a  great  number.  In  the  cluster  known  as  "  Messier  3 
(in  Canes  Venatici),"  132  have  been  detected ;  in  o>  Centauri,  122 ; 
in  Messier  5  (in  Libra),  85 ;  and  in  a  cluster  known  as  "  K  G.  C. 
7078,"  51.  In  all  five  hundred  and  fourteen  have  been  found  by  Mr. 
Bailey  in  globular  clusters.  The  periods  are  not  yet  fully  deter- 
mined, but  the  changes  are  very  rapid,  so  that  two  photographs  of 
Messier  5  taken  only  two  hours  apart  show  a  dozen  cases  in  which  the 
variation  of  brightness  amounts  to  a  full  magnitude. 

STAR   SPECTRA. 

(If  this  book  were  to  be  written  de  novo,  the  sections  upon  stellar  spectra 
would  be  placed  almost  at  the  beginning  of  Chapter  XX.,  since  in  dealing 
with  the  star-motions,  and  the  peculiarities  of  variable  stars,  it  is  now  con- 
tinually necessary  to  refer  to  spectroscopic  phenomena,  most  of  which  have 
been  discovered,  or  become  practically  important,  since  the  first  preparation 
of  the  work  in  1888.) 

855.  In  1824  Fraunhofer,  in  connection  with  his  study  of  the 
lines  of  the  solar  spectrum,  investigated  also  the  spectra  of  certain 

1  See  Addendum  C,  following  page  580. 


OBSERVATIONS    OF    HUGGINS    AND    SECCHI.  531 

stars,  using  an  apparatus  essentially  similar  to  that  which  is  now 
employed  at  Cambridge.  He  placed  a  prism  in  front  of  the  object- 
glass  of  a  small  telescope  and  looked  at  the  stars  through  this,  using 
a  cylindrical  lens  in  the  eye-piece  to  widen  the  spectrum,  which 
otherwise  would  be  a  mere  line. 

He  found  that  Sirius,  Castor,  and  many  other  stars  show  very  few  dark 
lines  in  their  spectrum,  but  strong  ones;  while,  on  the  other  hand,  the 
spectra  of  Pollux  (/3  Geminorum)  and  Capella  resemble  closely  the  spectrum 
of  the  sun.  In  all  the  spectra  he  recognized  the  D  line,  although  it  was 
not  then  known  that  it  had  anything  to  do  with  sodium. 

856.  Observations  of  Huggins  and  Secchi, — Almost  as  soon  as 
the  spectroscope  had  taken  its  place  as  a  recognized  instrument  of 
science  it  was  applied  to  the  study  of  the  stars  by  Kutherfurd  and 
Huggins,  and  Secchi  followed  hard  in  their  footsteps.     Huggins 
(now  Sir  William)  studied  the  spectra  of  comparatively  few  stars, 
but  with  all  the  dispersive  power  he  could  obtain,  and  in  detail ; 
while   Secchi,  using  a  much  less  powerful  instrument,  examined 
several  thousand  star  spectra,  in  a  more  general  way,  for  purposes 
of  classification. 

Huggins  identified  with  considerable  certainty  in  the  spectra  of 
a  Orionis  (Betelgeuze)  and  a  Tauri  (Aldebaran)  a  number  of  elements 
that  are  familiar  on  the  earth,  and  are  most  of  them  prominent  in 
the  solar  spectrum.  In  the  former  he  reported  sodium,  magnesium, 
calcium,  iron,  bismuth,  and  hydrogen ;  and  in  a  Tauri,  in  addition, 
tellurium,  antimony,  and  mercury;  but  these  latter  metals  have  not 
yet  been  verified. 

857,  Classification  of  Stellar  Spectra.  —  Secchi,  in  his  spectro- 
scopic  survey,  found  that  the  4000  stars  which  he  observed  could 
all  be  reduced  to  four  classes,  and  although  his  classification  can 
now  be  regarded  as  provisional  only,  and  by  no  means  complete  or 
satisfactory,  yet  it  is  more  generally  used  than  any  other,  either 
in  its  simple  form,  or  with  subdivisions  and  modifications  such  as 
Vogel,  Pickering,  and  others  have  introduced. 

According  to  Secchi :  — 

The  first  class  comprises  the  white  or  blue  stars.  To  it  belong  Sirius 
and  Vega,  and,  in  fact,  considerably  more  than  half  of  all  the  stars 
examined.  The  spectrum  is  characterized  by  the  great  strength  of 
the  hydrogen  lines,  which  are  wide,  hazy  bands,  much  like  the  H 


532 


CLASSIFICATION   OF    STELLAR   SPECTRA. 


and  K  lines  in  the  solar  spectrum.  Other  lines  are  extremely  faint 
or  entirely  absent ;  the  K 1  line  especially,  which  in  the  solar  spec- 
trum is  especially  prominent,  in  the  spectra  of  most  of  these  stars 
is  hardly  visible. 

The  second  class  is  also  numerous,  and  is  composed  of  stars  with  a 
spectrum  substantially  like  that  of  our  sun.  The  H  and  K  lines  are 
both  strong.  Capella  and  Pollux  (ft  Geminorum)  are  prominent 
examples  of  this  class.  There  are  certain  stars  which  form  a  con- 
necting link  between  these  two  first  classes,  stars  like  Procyon  and 
a  Aquilse,  which,  while  they  show  the  hydrogen  lines  very  strongly, 
also  exhibit  a  great  number  of  other  lines  between  them.  The  first 
and  second  classes  together  embrace  fully  seven-eighths  of  all  the 
stars  he  observed. 


FIG.  223.  —  Secchi'8  Types  of  Stellar  Spectra. 

The  third  class  includes  most  of  the  red  and  variable  stars,  some 
500  in  number,  and  the  spectrum  is  characterized  by  dark  bands 
instead  of  lines  (though  lines  are  generally  present  also).  These 

1  In  stars  of  this  class  the  "  H  line  "  in  their  spectra  is  not  the  H  line  of  cal- 
cium, but  the  Epsilon  line  of  hydrogen  (-He),  which  lies  close  to  the  calcium 
line,  as  shown  in  Fig.  116*,  Art.  326.  In  stars  of  the  second  or  "solar"  class, 
on  the  contrary,  both  H  and  K  are  practically  due  to  calcium  alone,  the  hydro- 
gen line  being  very  inconspicuous. 


STAR    SPECTRA.  533 

bands,  not  yet  identified  as  to  origin,  shade  from  the  blue  towards 
the  red;  that  is,  they  are  sharply  defined  and  darkest  at  the  more 
refrangible  edge.  Occasionally  in  spectra  of  this  type  some  of  the 
hydrogen  lines  are  bright,  a  Herculis,  a  Orionis,  and  Mir  a  (o  Ceti) 
are  fine  examples  of  this  third  class. 

The  fourth  class  is  composed  of  a  very  small  number  of  stars,  less 
than  sixty  so  far  as  known,  mostly  small  red  stars.  This  spectrum 
is  also  a  banded  one ;  but  compared  with  the  third  class  the  bands 
(probably  due  to  carbon)  are  reversed,  that  is,  are  shaded  towards  the 
blue.  These  generally  show  also  a  number  of  bright  lines.  None 
of  the  conspicuous  stars  belong  to  this  class  —  none  above  the  fifth 
magnitude.  The  sixth  magnitude  star,  152  Schjellerup,  may  be 
taken  as  its  finest  example  (a,  12h  40m;  8,  +45°  59',  in  the  constella- 
tion of  Canes  Venatici).  Fig.  223  exhibits  the  light-curves  of  these 
four  types  of  spectrum.1 

There  are  many  stars  which  seem  in  their  spectroscopic  characteristics  to 
lie  between  classes  1  and  2 ;  and  there  are  others  which  cannot  be  said  to 
belong  to  either  of  the  four.  Pickering  has  proposed  a  fifth  class,  mainly 
to  include  a  small  group,  known  as  the  "  Wolf-Rayet  stars  "  with  a  very 
peculiar  spectrum  of  bands  and  bright  lines.  Nearly  seventy  are  known  at 
present,  all  faint,  and  all  in,  or  near,  the  milky  way  and  Magellanic  clouds. 
They  seem  to  hold  an  important  place  in  the  still  obscure  theory  of  stellar 
development. 

858.     Vogel   has   revised    Secchi's   classification   of    spectra   as 
follows,  making  only  three  main  classes,  but  with  subdivisions: 
I.     (a)  Same  as  Secchi's  I.     The  white  stars. 

(£)  Nearly  continuous ;    all  lines  wanting  or  very  faint. 

(3  Orionis  is  the  type, 
(c)   Showing  the  lines  of   hydrogen  bright,  and  also  the 

helium  line  D3  (Art.  323). 
II.     (a)  Same  as  Secchi's  II. 

1  It  is  difficult  to  represent  spectra  accurately  by  any  process  of  engraving  that 
can  be  readily  reproduced  in  a  book  like  the  present.  The  curve,  on  the  other 
hand,  is  easily  managed,  and,  though  it  does  not  please  the  eye  like  the  spectrum 
itself,  it  is  capable  of  conveying  all  the  information  that  could  be  obtained  from 
the  most  finished  engraving.  Dark  lines  are  represented  by  lines  running  down- 
ward from  the  upper  boundary  line  of  the  curve,  and  bright  lines  by  lines  running 
upward,  while  the  bands  and  their  shading  are  represented  by  variations  in  the 
contour  of  the  curve. 


534  PHOTOGRAPHY    OF    STELLAR    SPECTRA. 

(b)   Like  II.  (a),  but  showing  bright  lines  which  are  not  the 
lines  of  hydrogen  or  helium.    (The  Wolf-Eayet  stars.) 

III.     (a)   Same  as  Secchi's  III. 
(b)   Same  as  Secchi's  IV. 

Vogel's  classification  is  based  in  part  on  the  very  doubtful  assumption 
that  stars  of  Class  I.  are  hottest  and  also  youngest,  while  the  other  classes 
belong  to  stars  which  are  either  beginning  to  fail  or  are  already  far  gone  in 
decrepitude.  But  it  is  very  far  from  certain  that  a  red  star  is  not  just  as 
likely  to  be  younger  than  a  white  one,  as  to  be  older.  It  probably  is  now 
at  a  lower  temperature,  and  possesses  a  more  extensive  envelope  of  gases;  but 
it  may  be  increasing  in  temperature  as  well  as  decreasing.  At  any  rate  we 
have  no  certain  knowledge  about  its  age. 

Since  the  identification  of  helium  and  its  numerous  lines  in  the  solar 
spectrum,  its  lines  have  also  been  recognized  in  the  spectra  of  many  stars 
(especially  numerous  in  Orion),  and  seem  likely  to  throw  much  light  on 
their  classification.  Vogel  has  taken  them  into  account  in  a  modified  form 
of  his  system,  which  we  have  not  space  to  present. 

Sir  Norman  Lockyer  has  also  proposed  a  very  elaborate  classification, 
based  on  his  "  Meteoritic  Theory  "  of  stellar-development  (see  Art.  926). 

859.  Photography  of  Stellar  Spectra.  —  As  early  as  1863  Huggins 
attempted  to  photograph  the  spectrum  of  Vega,  and  succeeded  in  getting  an 
impression  of  the  spectrum,  but  without  any  of  the  lines.     In  1872  Dr. 
Henry  Draper  of  New  York,  working  with  the  reflector  which  he  had  him- 
self constructed,  succeeded  in  getting  an  impression  of  the  spectrum  of  the 
same  star,  showing  for  the  first  time  four  of  its  hydrogen  lines.    The  intro- 
duction of  the  more  sensitive  dry  plates  in  1876  induced  Mr.  Huggins  to 
resume  the  subject  (as  did  Dr.  Draper  soon  after),  and  they  soon  succeeded 
in  getting  pictures  showing  many  lines.     The  spectra  were  about  half  an 
inch  long  by  T\  or  ^  of  an  inch  wide.     After  the  lamented  death  of  Dr. 
Draper  in  1882,  Professor  Pickering  took  up  the  work  at  Cambridge  (U.  S.); 
and  with  such  success  that  Mrs.  Draper,  who  had  intended  to  establish  and 
to  endow  her  husband's  observatory  as  an  establishment  for  astro-physical 
research,  and  a  most  fitting  monument  to  his  memory,  concluded  to  transfer 
the  instruments  to  Cambridge,  and  there  establish  the  "  Draper  Memorial," 
which  has  already  accomplished  so  much  for  spectroscopic  astronomy.    Other 
observers  have  followed  on,  both  in  Europe  and  in  this  country;  and  it  is 
hardly  too  much  to  say  that  four-fifths  of  all  stellar  spectroscopic  work  is  now 
done  by  photography.  Spectra  too  faint  to  be  even  seen  by  the  eye  can  be  photo- 
graphed, and  so  studied  in  their  minutest  peculiarities;  and  with  the  continual 
improvement  of  our  plates  the  range  of  possible  observation  increases  daily. 

860.  The  Slitless  Spectroscope.  —  Professor  Pickering  has  at- 
tained his  remarkable  success  by  reverting  to  the  "  slitless  spectro- 


THE    SLITLESS    SPECTROSCOPE. 


535 


scope, "  arranged  in  the  manner  first  used  by  Fraunhofer,  and  later 
revived  by  Secchi.  The  instrument  consists  of  a  telescope  with  the 
objective  corrected  not  for  the  visual,  but  for  the  photographic  rays, 
equatorially  mounted  and  carrying  in  front  of  the  object-glass  one  or 
more  "  objective-prisms  "  with  a  refracting  angle  of  from  10°  to  30°, 
and  large  enough  to  cover  the  whole  lens. 

The  refracting  edge  of  the  prism  is  placed  east  and  west,  so  that  the 
linear  spectrum  of  a  star  formed  on  a  plate  at  the  focus  of  the  object-glass 
runs  north  and  south.  If,  now,  the  clock-work  of  the  instrument  is  adjusted 
to  follow  the  star  exactly,  the  image  (I.e.,  the  spectrum)  will  be  a  mere  line, 
broken  here  and  there  where  the  dark  lines  of  the  spectrum  should  appear. 
By  merely  retarding  or  accelerating  the  clock  a  trifle,  the  linear  spectrum 
will  drift  a  little  sidewise  upon  the  plate,  and  so  will  form  a  spectrum  having 
a  width  depending  on  the  amount  of  this  drift  during  the  time  of  exposure. 
If  the  air  is  calm  the  lines  of  the  spectrum  thus  formed  are  as  clean  and 
sharp  as  if  a  slit  were  used;  otherwise  not. 

861.  The  instrument  hitherto  most  used  in  this  work  at 
Cambridge  is  Dr.  Draper's  eleven-inch  photographic  refractor, 
with  four  huge  glass  prisms  in  a  box  in  front  of  the  object- 
glass,  arranged  as  indicated 
in  Fig.  224.  With  this 
apparatus,  photographic 
spectra  of  the  brighter 
stars  are  now  obtained  hav- 
ing, before  enlargement,  a 
length  of  fully  three  inches 
from  F  in  the  blue  of  the 
spectrum  to  the  extremity 
of  the  ultra  violet.  It  is 
a  pity,  of  course,  that  the 
lower  portions  of  the  spec- 
trum below  F  cannot  be 
reached  in  the  same  way; 

but  no  plates  sufficiently  sensitive  to  green,  yellow,  and  red  rays  have 
yet  been  found.  The  exposure  necessary  to  obtain  the  impression  of 
even  the  most  powerful  photographic  rays  is  from  half  an  hour  to  an 
hour.  Fig.  225  is  enlarged  about  one-third  from  one  of  these  photo- 
graphs of  the  spectrum  of  Vega,  which  extends  far  into  the  ultraviolet. 

At  the  extreme  left,  the  figure  fails  to  show  properly  the  perfect  regularity 
with  which  the  hydrogen  lines  crowd  closer  and  closer  together,  in  exact 


FIG.  224. 
Arrangement  of  the  "  objective-prisms. 


536  THE    SLITLESS    SPECTROSCOPE. 

accordance  with  a  remarkable  formula  discovered  by  Balmer  in  1885,  viz., 

a 

\  =  X0 — -,  in  which  m  takes  successively  the  values  3,  4,  5,  etc.,  \0 

being  3646.1.     (See  note  at  end  of  the  chapter,  p.  538.) 

These  spectra  bear  tenfold  enlargement  perfectly,  making  them 
more  than  two  feet  long  by  two  inches  in  width,  and  then  in  the 
spectrum  of  such  a  star  as  Capella  they  show  hundreds  of  lines. 
It  is  simply  amazing  that  the  feeble,  twinkling  light  of  a  star  can 
be  made  to  produce  such  an  autographic  record  of  the  substance 
and  condition  of  the  inconceivably  distant  luminary. 

Several  other  photographic  telescopes  of  much  larger  size,  both  in  Europe 
and  in  America,  are  now  fitted  with  objective-prisms.  The  Bruce  telescope 
of  24  inches  aperture,  which  has  already  been  mentioned  (Art.  798*),  is  at 


ft  fl  H  ry 

FIG.  225.  —  Photographic  Spectrum  of  Vega.    Cambridge,  1887. 

present  the  largest.  Theoretically,  a  diffraction  grating  would  answer  the 
same  purpose  as  a  prism,  and  some  experiments  have  been  made  as  to  the 
practicability  of  constructing  such  a  grating  for  use  with  the  great  Yerkes 
telescope,  by  following  Fraunhofer's  original  plan  of  winding  fine  wires  upon 
a  pair  of  finely  threaded  screws.  Promising  experiments  are  also  being 
made  with  concave  gratings. 

In  still  another  form  of  instrument,  the  spectroscope  is  attached  to  the 
eye  end  of  the  telescope  just  as  usual,  the  slit  of  the  collimator  being  simply 
omitted. 

862.     Peculiar  Advantages  of  the  Slitless  Spectroscope.  —  The 

slitless  spectroscope  has  three  great  advantages.  First,  that  it 
utilizes  all  the  light  that  comes  from  the  star  to  the  object-glass, 
much  of  which  in  the  usual  form  of  the  instrument  is  lost  in  the 
jaws  of  the  slit.  Secondly,  that  by  taking  advantage  of  the  length 
of  a  large  telescope,  it  produces  a  very  high  dispersion  with  even  a 
single  prism.  Thirdly,  and  most  important,  it  gives  on  the  same 
plate  and  with  a  single  exposure  the  spectra  of  all  the  many  stars 
whose  images  fall  upon  it.  With  the  smaller  eight-inch  instrument 
made  at  Cambridge,  and  one  prism,  as  many  as  100  or  150  spectra 
are  sometimes  taken  together;  as,  for  instance,  in  a  spectrum  photo- 


THE    SLITLESS    SPECTROSCOPE.  537 

graph  of  the  Pleiades.  For  purposes  of  "  reconnoissance,"  there- 
fore, where  the  object  is  to  obtain,  compare,  and  classify  the  spectra 
of  thousands  of  stars,  this  form  of  instrument  is  unrivalled. 

863,  Disadvantages  of  the  Slitless  Spectroscope. — Per  contra, 
the  giving  up  of  the  slit  precludes  all  the  usual  methods  of  identify- 
ing the  lines  by  actually  confronting  them  with  comparison  spectra; 
the  comparison  prism  (Art.  315)  cannot  be  used.      This  makes  it 
extremely  difficult  to  utilize  these  magnificent  pictures  for  purposes 
of  scientific  measurement. 

Several  methods  have  been  proposed  by  which,  theoretically,  the  difficulty 
might  be  overcome;  none  of  them,  however,  offer  any  practical  approach  to 
the  accuracy  obtainable  with  slit-spectroscopes,  which  up  to  the  present  time 
have  been  exclusively  used  by  Vogel,  Frost,  and  Campbell  in  all  absolute 
measurements  of  motion  in  the  line  of  sight,  all  determinations  of  wave- 
length, and  all  trustworthy  identifications  of  stellar  elements.  In  certain 
cases  of  relative  motion,  however  (Art.  879),  objective-prism  spectroscopes 
have  already  done  good  work,  and  Professor  Pickering  has  very  lately  (1896) 
devised  a  most  ingenious  method  of  extending  their  range  to  a  determination 
of  the  motions  of  all  the  stars  of  a  group  relative  to  some  one  selected  as  a 
reference  point.  Two  photographs  of  the  spectra  are  made,  one  with  the 
telescope  on  one  side  of  the  pier,  and  the  red  ends  of  all  the  spectra  towards 
the  north,  say.  Then  the  telescope  is  reversed  to  the  other  side  of  the  pier, 
and  a  second  negative  is  made  in  which  the  spectra  will  have  their  red  ends 
to  the  south.  Moreover,  this  second  negative  is  made  with  the  sensitive  film 
turned  away  from  the  object-glass.  On  putting  the  two  plates  together,  with 
films  next  each  other,  and  making  the  two  spectra  of  the  "  guide-star  "  coin- 
cide, all  the  other  spectra  will  also  coincide ;  —  exactly,  in  cases  where  the 
"  radial-motion  "  of  the  stars  is  the  same  as  that  of  the  guide-star  :  otherwise 
there  will  be  a  slight  want  of  coincidence  in  the  lines  that  ought  to  agree, 
and  half  this  difference  of  position  will  be  the  "  displacement "  of  the  spec- 
trum lines  due  to  the  difference  between  the  radial  velocity  of  the  guide-star 
and  that  of  the  star  in  question.  The  method  has  been  tried  upon  the 
Pleiades,  but,  rather  disappointingly,  did  not  bring  out  any  evidence  of  a 
general  rotation  of  the  cluster  around  its  centre. 

864.  Twinkling  or  Scintillation  of  the  Stars.  —  This  is  a  purely 
atmospheric  effect,  usually  violent  near  the  horizon  and  almost  null 
at  the  zenith.    It  differs  greatly  on  different  nights  according  to  the 
steadiness  of  the  air. 

If  the  spectrum  of  a  star  near  the  eastern  horizon  be  examined 
with  a  spectroscope  so  held  as  to  make  the  spectrum  vertical,  it  will 


538  CAUSE   OF   SCINTILLATION. 

appear  to  be  continually  traversed  by  dark  bands  running  through, 
the  spectrum  from  the  blue  end  towards  the  red.  At  the  western 
horizon  the  bands  move  in  the  opposite  direction,  from  red  to  blue; 
on  the  meridian  they  merely  oscillate  back  and  forth. 

Cause  of  Scintillation.  —  Authorities  differ  as  to  the  exact  explanation 
of  scintillation,  but  probably  it  is  mainly  due  to  two  causes  (optically  speak- 
ing), both  depending  on  the  fact  that  the  air  is  full  of  streaks  of  unequal 
density  that  are  carried  by  the  wind. 

(1)  In  the  first  place,  light  transmitted  through  such  a  medium  is  con- 
centrated in  some  places  and  turned  away  from  others  by  simple  refraction : 
so  that,  if  the  light  of  a  star  were  strong  enough,  a  white  surface  illuminated 
by  it  would  look  like  the  sandy  bottom  of  a  shallow,  rippling  pool  of  watei 
illuminated  by  sunlight,  with  light  and  dark  mottlings  which  move  with  the 
ripples  on  the  surface.     So,  as  we  look  towards  the  star,  and  the  mottlings 
due  to  the  irregularities  of  the  air  move  by  us,  we  see  the  star  alternately 
bright  and  faint;  in  other  words,  it  twinkles;  and  if  we  look  at  it  in  a 
telescope  we  shall  see  that  it  not  only  twinkles,  but  dances,  i.e.,  it  is  slightly 
displaced  back  and  forth  by  the  refraction. 

(2)  The  other  cause  of  twinkling  is   "  interference."     Pencils   of  light 
coming  from  the  star  (which  optically  is  a  mere  point),  and  feebly  refracted 
by  the  air  in  the  way  above  explained,  reach  the  observer  by  slightly  differ- 
ent routes,  and  are  just  in  a  condition  to  interfere.      The  result  of  the 
interference  is  the  temporary  destruction  of  rays  of  certain  wave-lengths, 
and  the  reinforcement  of  others.     At  a  given  moment  the  green  rays,  for 
instance,  will  be  destroyed,  while  the  red  and  blue  will  be   abnormally 
intense;  hence  the  quivering  dark  bands  in  the  spectrum.    If  the  star  is  very 
near  the  horizon,  the  effects  are  often  sufficient  to  produce  marked  changes 
of  color. 

865.  Why  Planets  Twinkle  Less  than  Stars.  —  This  is  mainly 
because  they  have  discs  of  sensible  diameter,  so  that  there  is  a  general 
unchanging  average  of  brightness  for  the  sum  total  of  all  the  lumi- 
nous points  of  which  the  disc  is  composed.  When,  for  instance, 
point  A  of  the  disc  becomes  dark  for  a  moment,  point  B,  very  near 
it,  is  just  as  likely  to  become  bright ;  the  interference  conditions 
being  different  for  the  two  points.  The  different  points  of  the  disc 
do  not  keep  step,  so  to  speak,  in  their  twinkling. 

865*.  NOTE  ON  "  SERIES  "  IN  SPECTRA.  —  In  1896  Professor  Pickering 
found  on  the  Draper  Memorial  photographs  a  remarkable  series  of  lines  in 
the  spectra  of  certain  stars,  of  which  f  Puppis  is  the  most  conspicuous. 
These  lines  fall  regularly  intermediate  between  the  lines  of  hydrogen,  and 


EXERCISES.  539 

he  soon  after  discovered  that  a  single  formula,  which  includes  that  of 
Balmer,  gives  the  positions  both  of  these  lines  and  those  of  hydrogen 

n2 

formerly  known:  it  is,  A.  =  3646.1  —    — .     If  in  this  we  give  to  n  the  even 

n2  —  16 

values,  6,  8,  10,  etc.,  we  get  the  wave-lengths  of  the  lines  of  hydrogen  as 
formerly  known:  the  odd  values  7,  9,  11,  etc.,  give  the  new  lines.  It  seems, 
therefore,  extremely  likely  that  these  new  lines  also  belong  to  hydrogen,  in 
some  condition,  however,  which  differs  from  that  which  obtains  in  our 
laboratories,  and  in  stars  like  Vega. 

The  investigations  of  Kayser  and  Runge  (1888-1896)  have  shown  that  the 
lines  in  the  spectra  of  many,  if  not  most,  of  the  elements  are  spaced  in  a 
somewhat  similar  manner,  expressible  by  a  simple  formula;  usually,  how- 
ever, two  or  more  distinct  series  are  found.  In  the  spectrum  of  helium, 
according  to  Runge,  there  are  two  "  sets,"  each  set  consisting  of  a  principal 
series,  and  two  subordinate  series,  —  six  in  all.  In  most  cases,  the  regular 
"  series  "  exclude  some  of  the  lines  that  appear  in  the  spectrum  of  a  given 
element,  and  not  unfrequently  these  independent  lines  are  among  the  most 
conspicuous  and  important  of  all.  Thus,  the  H  and  K  lines  do  not  belong 
to  either  of  the  two  regular  series  in  the  spectrum  of  calcium.  The  explana- 
tion of  these  series  is  not  yet  known,  but  it  probably  depends  somehow  upon 
the  manner  in  which  the  atoms  are  arranged  in  the  molecule. 


EXERCISES  ON  CHAPTER  XXI. 

1.  What  is  the  brightness  of  a  star  of  the  10.5  magnitude  (on  the  abso- 
lute scale)  compared  with  that  of  a  star  of  the  standard  first  magnitude  ? 

From  Art.  820  we  have  log  610.5=  log  6t  —  T%  x  9.5.  If  we  take  the  brightness  of  the  first 
magnitude  star  as  the  unit  of  brightness  log  6t=  0,  and  we  have  log  610.5  =  0  —  0.4  x  9.5=  — 
3.8000.  To  bring  this  entirely  negative  logarithm  into  the  usual  tabular  form,  in  which  the 
characteristic  only  is  negative  while  the  mantissa  is  positive,  we  numerically  increase  the 
characteristic  by  unity,  making  it  —  4,  and  at  the  same  time  take  for  the  new  mantissa  1  — 
0.8000,  or  .2000  ;  we  have,  therefore,  log  bw.s=  4.2000  ;  whence,  from  the  logarithmic  table,  we 
find  6,0.5=  0.000158. 

Also  log  ^~  =  0  -  (  -  3.8000)  =  +  3.8000  ;  whence,  b^  =  6309.6  x  610>5. 

"10-5 
(In  all  computations  respecting  stellar  magnitudes  four-place  tables  are  sufficient.) 

2.  What  is  the  brightness  of  an  eleventh  magnitude  star  in  terms  of  the 
first? 


Ans.    0.0001,  or 
I/ 
3.    What  is  the  brightness  of  a  4.8  magnitude  star  in  terms  of  the  first? 

Ans.    0.0302,  or-- 


540  EXERCISES. 

\ 

4.  What  is  the  magnitude  of  a  star  whose  brightness  is  one  one-hundred- 
thousandth  that  of  a  first  magnitude  star?     (Art.  820,  Eq.  2.) 

Am.    13.5  magnitude. 

5.  What  is  the  magnitude  of  a  star  a  millionth  as  bright  as  a  first 
^gnitude?  An,.    16th  magnitude. 

*•  6.  WThat  is  the  magnitude,  on  the  absolute  scale,  of  a  luminary  80000- 
000000  times  as  bright  as  a  first  magnitude  star?  (Log  80000000000  = 
10.9031.)  Ans  _  26.26  magnitude. 

(This  is  about  the  estimated  brightness  of  the  sun.) 

'  7.    What  is  the  apparent  magnitude  of  a  double  star  whose  components 
are  of  the  first  and  second  magnitudes  respectively? 

Ans.    0.64  magnitude. 

y 

8.    What,  if  the  components  are  of  the  second  and  fourth  magnitudes? 

Ans.    1.85  magnitude. 

/9.  If  the  distance  of  a  fourth  magnitude  star  were  diminished  one-half, 
of  what  magnitude  would  it  appear  ? 

Ans.    2.50  magnitude. 

J 

10.  If  the  distance  of  a  star  were  increased  by  forty  per  cent,  how  much 
would  its  magnitude  be  changed  ? 

Ans.    0.73  of  a  magnitude,  numerical  increase. 
\J 

11.  If  the  distance  of  a  star  were  diminished  by  forty  per  cent,  how 
would  its  magnitude  be  affected? 

Ans.    1.11  of  a  magnitude,  numerical  decrease. 

*   12.   If  a  star  of  the  9th  magnitude  has  a  parallax  of  0.25",  how  does  the 
light  emitted  by  it  compare  with  that  of  the  sun? 

Ans.    Tfo. 

13.    With  the  data  given  in  Table  IV.  compute  the  light-emission  of  other 
stars  compared  with  that  of  the  sun. 


DOUBLE    AND    MULTIPLE    STARS. 


541 


CHAPTER  XXII. 


DOUBLE     AND     MULTIPLE     STARS.  —  ORBITS     AND     MASSES     OF 

DOUBLE  STARS. CLUSTERS. —  NEBULAE. THE  MILKY  WAY. 

-  DISTRIBUTION  OF  STARS. —  CONSTITUTION  OF  THE  STELLAR 
UNIVERSE. —  COSMOGONY  AND  THE  NEBULAR  HYPOTHESIS. 

866.  Double  and  Multiple  Stars,  —  The  telescope  shows  numer- 
ous instances  in  which  two  stars  lie  very  near  each  other,  in  many 
cases  so  near  that  they  can  be  seen  separate  only  under  a  high 


FIG.  226.  — Double  and  Multiple  Stars. 


magnifying  power.  These  are  called  "double  stars."  At  present 
no  less  than  16000  such  couples  are  known,  and  the  number  is 
continually  increasing.  In  not  a  few  instances  we  have  three  stars 


542  DISTANCE,    MAGNITUDES,    AND    COLORS. 

together, .  two  of  which,  are  usually  very  close  and  the  third  farther 
away;  and  there  are  several  cases  of  quadruple  stars,  where  there 
are  two  pairs  of  stars  lying  close  together  (as  in  €  Lyrse),  or  a  pair 
of  stars  with  two  single  stars  close  by;  and  there  are  some  cases 
where  more  than  four  form  a  "  multiple  star."  Fig.  226  represents 
a  number  of  such  double  and  multiple  stars. 

867.  Distance,  Magnitudes,  and  Colors. —  The  apparent  distances 
usually  range  from  30"  to  £",  few  telescopes  being  able  to  separate 
double  stars  closer  than  £". 

In  a  very  large  proportion  of  cases  (perhaps  about  one-third  of  all) 
the  two  stars  are  nearly  equal ;  in  many  others  they  are  extremely 
unequal,  a  minute  star  near  a  large  one  being  usually  known  as  its 
"  companion." 

Not  infrequently  the  components  of  a  double  star  present  a  fine 
contrast  of  color;  never,  however,  in  cases  where  they  are  nearly  equal 
in  magnitude.  It  is  a  remarkable  fact,  as  yet  wholly  unexplained, 
that  when  we  have  such  a  contrast  of  color  the  tint  of  the  smaller 
star  always  lies  higher  in  the  spectrum  than  that  of  the  larger  one. 
The  larger  one  is  reddish  or  yellowish,  and  the  smaller  one  green  or 

blue,  without  a  single  ex- 
ception among  the  many 
hundreds  of  such  tinted 
couples  now  known,  y 
Andromedse  and  /?  Cygni 
are  fine  examples  for  a 
small  telescope. 

868.  Measurement  of 
Double  Stars. — Such  meas- 
ures are  generally  made 
with  a  filar  position-mi- 
crometer, essentially  such 
as  shown  in  Figs.  28  and 
29  (Art.  73).  The  quan- 
tities to  be  determined  are 

the  distance  and  position- 
Measurement  of  Dteta^£  and^Position-Angle  of  a         angle  Q£  the  CQUple>      By 

"distance"    we    mean 

simply  the  apparent  distance  in  seconds  of  arc  between  the  centres 
of  the  two  star  dies.  The  position-angle  of  a  double  star  is  the  angle 


STABS    OPTICALLY   AND   PHYSICALLY   DOUBLE.  543 

made  with  the  hour-circle  by  the  line  drawn  from  the  larger  star  to  the 
smaller,  reckoning  around  from  the  north  through  the  east,  as  shown 
in  Fig.  227. 

Photography  may  also  be  used,  and  promises  to  become  very  useful  in 
cases  where  the  distance  is  not  less  than  4"  or  5". 

869.  Stars   Optically  and   Physically  Double. —  Stars  may  be 
double  in  two  different  ways.    They  may  be  merely  optically  double, 

—  that  is,  simply  in  line  with  each  other,  but  one  far  beyond 
the  other;  or  they  may  be  really  very  near  together,  in  which 
case  they  are  said  to  be  "physically  connected"  because  they  are 
then  under  the  influence  of  their  mutual  attraction,  and  move 
accordingly. 

870.  Criterion  for  distinguishing  between  Physically  and  Optic- 
ally Double  Stars. —  This  cannot  be  done  off-hand.     It  requires  a 
series  of  measurements  long  enough  continued  to  determine  whether 
the  relative  movement  of  the  stars  is  in  a  curve  or  a  straight  line.    If 
the  stars  are  really  close  together  their  attraction  will  force  them 
to  describe  curves  around  each  other.     If  they  are  really  at  a  great 
distance  and  only  accidentally  in  line,  then  their  proper  motions, 
being   sensibly   uniform   and   rectilinear,   will   produce   a  relative 
motion  of  the  same  kind.     Taking  either  star  as  fixed,  the  other 
star  will  appear  to  pass  it  in  a  straight  line,  and  with  a  steady, 
uniform  drift. 

871.  Relative  Number  of  Stars  Optically  Double  and  Physically 
Connected.  —  Double-star  observations   practically  began  with   Sir 
William  Herschel  only  a  little  more  than  a  hundred  years  ago. 
When  he  took  up  the  subject  less  than  100  such  pairs  had  been 
recognized,  such  as  had  been  accidentally  encountered  in  making 
observations  of  various  kinds.     The  great  majority  of  double  stars 
have  been  discovered  so  recently  that  sufficient  time  has  not  yet 
elapsed  to  make  the  criterion  above  given  effective  with  more  than 
a  small  proportion  of  them.     But  it  is  already  perfectly  clear  that 
the  optically  double  stars  are,  as  the  theory  of  probability  shows 
they  ought  to  be,  very  few  in  number,  while  several  hundred  pairs 
have  shown  themselves  to  be  physically  connected,  i.e.,  to  be  what 
are  known  as  "  binary  "  stars,  or  couples  which  revolve  around  their 
common  centre  of  gravity. 


544  BINARY   STARS. 

872.  Binary  Stars.  —  Sir  W.  Herschel  began  his  observations  of 
double  stars  in  the  hope  of  ascertaining  stellar  parallax.     He  had 
supposed  in  the  case  of  couples  where  one  was  large  and  the  other 
small  that  the  smaller  one  was  usually  a  long  way  beyond  the  other 
(as  sometimes  is  really  the  fact).     In  this  case  there  should  be  per- 
ceptible variations  in  the  distance  and  position  of  the  two  stars  dur- 
ing the  course  of  the  year ;  precisely  such  variations  as  those  by 
which,  fifty  years  later,  Bessel  succeeded  in  getting  the  parallax  of 
61  Cygni  (Art.  811).     But  Herschel,  instead  of  finding  the  yearly 
oscillation  of  distance  and  position  which  he  expected,  found  quite 
a  different  and,  at  the  time,  a  surprising  thing,  — a  regular,  progres- 
sive change,  which  showed  that  one  of  the  stars  was  slowly  describ- 
ing a  regular  orbit  around  the  other.     To  use  his  own  expression, 
he  "  went  out  like  Saul  to  seek  his  father's  asses,  and  found  a  king- 
dom/7 — the  dominion  of  gravitation1  extended  to  the  stars,  unlimited 
by  the  bounds  of  the  solar  system,      y  Virginis,  £  Ursae  Majoris, 
and  £  Herculis  were  among  the  most  prominent  of  the  systems 
which  he  pointed  out. 

At  present  the  number  of  pairs  supposed  to  be  binary  is  at  least 
300,  and  as  many  more  begin  to  show  signs  of  movement.  (Up 
to  the  present  time  of  course  only  the  quicker  moving  ones  are 
obvious).  About  ninety  have  progressed  so  far,  —  having  made  at 
least  one  entire  revolution  or  a  great  part  of  one, — that  their  orbits 
have  been  computed  more  or  less  satisfactorily. 

873.  Orbits  of  Binary  Stars. — The  real  orbit  described  by  each 
of  such  a  pair  of  stars  is  always  found  to  be  an  ellipse,  and  assum- 
ing the  applicability  of  the  law  of  gravitation,  the  common  centre 
of  gravity  must  be  at  the  focus.     The  two  ellipses  are  precisely 
similar,  the  one  described  by  the  smaller  star  being  larger  than  the 
other  in  inverse  proportion  to  the  star's  mass. 

So  far  as  the  relative  motion  of  the  two  bodies  goes,  we  may  regard 
either  of  them  (usually  the  larger  is  preferred)  as  being  at  rest,  and 
the  other  as  moving  around  it  in  a  relative  orbit  of  precisely  the  same 
shape  as  either  of  the  two  actual  orbits  which  are  described  around 
the  centre  of  gravity.  But  the  relative  orbit  is  larger,  having  for 

1  It  is  not  yet  fully  demonstrated  that  the  motions  of  binary  stars  are  due  to 
gravitation,  though  it  is  extremely  probable,  and  the  burden  of  proof  seems  to 
be  shifted  upon  those  who  are  disposed  to  doubt  it.  See,  however,  the  foot-note 
to  Art.  901. 


CALCULATION   OF    THE    ORBIT    OF    A   BINABY    STAR.      545 

its  semi-major  axis  the  sum  of  the  two  semi-axes  of  the  real  orbits 
(Art.  427). 

Usually  the  relative  orbit  is  all  that  we  can  ascertain  at  present, 
as  this  alone  can  be  deduced  from  the  micrometer  measures  when 
they  consist  only  of  position-angles  and  distances  measured  between 
the  two  stars.  / 

In  a  few  cases  where  such  measures  have  been  made  from  small  stars  in 
the  same  field  of  view  with  the  couple,  but  not  belonging  to  the  system,  or 
when  the  couple  has  long  been  observed  with  the  meridian  circle,  it  becomes 
possible  to  work  out  separately  the  apparent  orbit  of  each  star  of  the  pair 
with  reference  to  their  common  centre  of  gravity.  It  will  also  be  possible 
ultimately  to  compute  the  actual  orbits  of  many  double  stars,  independent  of 
any  hypothesis,  by  help  of  their  radial  motions  spectroscopically  determined 
in  different  parts  of  the  orbit.  But  this  can  be  done  only  in  cases  where 
the  components  are  not  too  close  to  permit  their  spectra  to  be  separately 
observed.  The  spectroscopic  and  micrometer  measures  combined  (if  strictly 
correct)  would  absolutely  determine  the  form,  size,  position  (and  distance 
also)  of  the  binary  system,  the  law  of  the  central  force,  and  the  masses  of 
the  component  stars. 

874.  Calculation  of  the  Orbit  of  a  Binary  Star.  —  If  the  observer 
is  so  placed  as  to  view  the  orbit  perpendicularly,  he  will  see  it  in  its 
true  form  and  having  the  larger  star  in  its  focus,  while  the  smaller 
moves  around  it,  describing  "  equal  areas  in  equal  times/'  But  if 
the  observer  is  anywhere  else,  the  orbit  will  be  apparently  more  or 
less  distorted.  It  will  still  be  an  ellipse  (since  every  projection  of 
a  conic  is  also  a  conic),  but  the  large  star  will  no  longer  occupy  its 
focus,  nor  will  the  major  and  minor  axes  be  apparently  at  right 
angles  to  each  other ;  nor  will  they  even  coincide  with  the  longest 
and  shortest  diameters  of  the  ellipse.  In  this  distorted  ellipse  the 
smaller  star  will,  however,  still  describe  equal  areas  in  equal  times 
around  the  larger  one. 

Theoretically,  five  absolutely  accurate  observations  of  the  position 
and  distance  are  sufficient  to  determine  the  elements  of  the  relative 
orbit,  if  we  assume  that  the  orbital  motion  is  described  under  the  law 
of  gravitation.  Practically  a  greater  number  are  needed  in  most 
cases,  because  the  motions  are  so  slow  and  the  stars  so  near  each 
other  that  observation-errors  of  0".l  (which  in  most  calculations  are 
of  small  account)  here  become  important.  The  work  requires  not 
only  labor,  but  judgment  and  skill,  and  unless  the  pair  has  completed 
or  nearly  completed  an  entire  revolution  the  result  is  apt  to  be  seri- 


546  SIRIUS   AND   PROCYON. 

ously  uncertain.  So  far,  as  has  been  said,  about  sixty  such  orbits 
are  fairly  well  determined.  Catalogues,  more  or  less  complete,  will 
be  found  in  Flaminarion's  book  on  "  Double  Stars,"  also  in  Gledhill's 
"Hand-book  of  Double  Stars,"  and  Houzeau's  "Vade  Mecum." 
Table  V.  in  the  Appendix  gives  the  elements  of  twenty-two  of  the 
best-known  orbits,  mostly  from  the  recent  calculations  of  Dr.  See. 

875.  SirillS  and  Procyon.  —  The  cases  of  these  two  stars  are  remark- 
able. In  both  instances  the  large  stars  have  been  found  from  meridian- 
circle  observations  to  be  slowly  moving  in  little  ellipses,  although  when  this 
discovery  was  first  made  neither  of  them  was  known  to  be  double.  In  1862 
the  minute  companion  of  Sirius  was  discovered  by  Clark  with  the  object- 
glass  of  the  Chicago  telescope,  then  just  finished,  and  at  that  time  the 
largest  object-glass  in  the  world.  And  this  little  companion  was  found  to  be 
precisely  the  object  needed  to  account  for  the  peculiar  motion  of  Sirius  itself. 


FIG.  228.  —  Orbits  of  Sirius  and  his  Companion. 

Fig.  228  represents  the  absolute  orbits  of  the  two  stars  and  also  the 
relative  orbit  of  the  smaller  star  around  the  larger  when  the  latter  is 
regarded  as  being  at  rest.  The  small  circle  at  C  is  the  earth's  orbit  drawn 
on  the  same  scale. 

In  the  case  of  Procyon,  the  companion  was  discovered  only  in  Nov.  1896, 
by  Schaeberle  at  the  Lick  Observatory  and  its  orbit  is  not  yet  worked  out. 

876.  Periods. —  The  periods  of  binary  stars,  so  far  as  at  present 
determined  by  micrometric  measures,  vary  from  5^  years  (the  period 
of  LI.  9091  according  to  See  :  Burnham  denies  it)  to  several  hundred 
years ;  though  none  of  the  very  long  periods  are  yet  accurately 
ascertained. 


SIZE   OF   THE   ORBITS. 


547 


It  is  possible  that  one  or  two  others  may  be  found  with  periods  even 
shorter  than  eleven  years,  and  it  is  practically  certain  that  as  time  goes  on, 
pairs  of  longer  period  than  1500 
years  will  present  themselves. 

Fig.  228*  shows  the  apparent 
orbits  of  several  of  the  most  in- 
teresting binaries. 


1848 


1718     1885    C 
J  Virginia . 


1827 


4.878 

500  ±  Years. 


1880   1{ 


.1750 


877.     Size  of  the  Orbits.  — 

The  angular  semi-major  axes 
of  the  orbits  thus  far  computed 
range  from  about  0".3  for  8 
Equulei,  to  18"  for  a  Centauri. 
The  real  dimensions  are,  of 
course,  only  to  be  obtained 
when  we  know  the  star's  par- 
allax and  distance.1  Fortu- 
nately several  of  the  stars 
whose  parallaxes  have  been  FIG.  223*.  —  Orbits  of  Binary  stars, 

determined  are  also  binary  stars.  Assuming  the  data  as  to  parallax 
and  orbits  given  in  the  tables  in  the  Appendix  we  find  the  following 
results :  — 


'180° 


ISO6 


-90° 


0° 
61  Cygni 


NAME. 

Assumed 
Parallax. 

Angular 
Semi-axis. 

Keal 
Semi-axis. 

Period. 

Mass. 
0=1. 

t]  Cassiopeiae    .... 
Sirius      

0".35 
0.39 

8".21 
803 

23.5 
20  6 

195y.8 
52  2 

0.33 
3  13 

a  Centauri  

0.75 

17  70 

23  6 

81  1 

2  00 

70  Ophiuchi     .... 

0.25 

4.54 

18.2 

88.4 

0.77 

In  the  case  of  Sirius,  the  companion  appears  to  have  a  mass  about 
four-tenths  that  of  the  principal  star,  while  the  two  components  of 
a  Centauri  are  very  nearly  equal  in  mass,  though  not  in  brightness. 

It  is  obvious  from  the  table  that  the  double-star  orbits  are  com- 
parable with  the  larger  orbits  of  the  solar  system,  all  four  of  them 
being  approximately  equal  to  that  of  Uranus.  Of  course  the  many 
binary  stars  whose  distance  is  so  great  as  to  make  their  parallax 
insensible  while  their  apparent  orbits  are  as  large  as  those  given  in 
the  list  must  have  real  orbits  of  still  vaster  dimensions. 

1  The  real  semi-axis  of  the  orbit  in  astronomical  units  is  simply  the  angular 
semi-axis  divided  by  the  parallax. 


548  MASSES    OF    BINA11Y    STARS. 

The  most  characteristic  peculiarity  of  the  double-star  orbits  is  their  large 
eccentricity,  which  for  the  sixty,  now  fairly  determined,  averages  very  nearly 
0.50.  Dr.  See  of  Chicago  explains  the  fact  on  the  hypothesis  that  the  binary 
stars  are  the  result  of  a  process  of  "tidal-evolution"  (Arts.  484  and  916). 
It  is  supposed  that  what  was  originally  a  nebulous  "  lump  "  assumed  a  dumb- 
bell form  in  whirling,  and  then  the  two  parts  separated,  still  revolving  around 
their  common  centre  of  gravity,  and  rotating  on  their  axes.  In  this  case 
the  tidal  reaction  between  the  two  would  be  extremely  powerful  and  effective, 
and  See,  following  very  nearly  the  methods  of  Darwin's  original  investiga- 
tion, has  shown  that  the  result  would  be  to  cause  them  to  move  apart,  and 
assume  orbits  more  and  more  eccentric  up  to  limits  depending  upon  the 
initial  conditions.  The  theory  is  now  very  generally  accepted  as  the  sub- 
stantial truth. 

878.  Masses  of  Binary  Stars.  —  When  we  know  both  the  size  of 
the  orbit  of  a  binary  and  its  period,  the  mass,  according  to  the  law 
of  gravitation,  follows  at  once  from  the  equation  of  Art.  536, 


If  t  and  a  are  given  respectively  in  years  and  astronomical  units 
of  distance,  then,  by  omitting  the  factor  4?r2,  M-\-  m  comes  out  in 
terms  of  the  sun's  mass.  The  final  column  of  the  little  table  above 
gives  the  masses  of  the  four  pairs  of  stars  as  compared  with  the 
mass  of  the  sun.  But  the  student  must  bear  in  mind  that  the  par- 
allaxes of  stars  are  so  uncertain  that  these  results  are  to  be  accepted 
with  a  very  large  margin  of  error. 

879.  Spectroscopic  Binaries,1  —  One  of  the  most  interesting  of 
recent  astronomical  results  is  the  detection  by  the  spectroscope  of 
several  pairs  of  double  stars  so  close  that  no  telescope  can  separate 
them.  In  1889  the  brighter  component  of  the  well-known  double 
star  Mizar  (Zeta  Ursse  Majoris,  Fig.  226)  was  found  by  Pickering  to 
show  the  dark  lines  double  in  the  photographs  of  its  spectrum,  at 
regular  intervals  of  about  fifty-two  days.  The  obvious  explanation 
is  that  this  star  is  composed  of  two,  which  revolve  around  their 
common  centre  of  gravity  in  an  orbit  which  is  turned  nearly  edge- 
wise to  us.  When,  twice  in  their  revolution,  the  line  that  joins  the 
two  stars  is  perpendicular  to  the  line  along  which  we  view  them, 
one  of  the  two  will  be  moving  towards  us,  while  the  other  is 
moving  in  an  opposite  direction;  and  as  a  consequence,  the  lines  in 
their  spectra  will  be  shifted  opposite  ways,  according  to  Doppler's 
principle.  Now  since  the  two  stars  are  so  close  that  their  spectra 

1  See  footnote  on  page  580. 


SPECTROSCOPIC   BINARIES.  549 

overlie  each  other,  the  result  will  be  simply  to  make  the  lines  in 
the  compound  spectrum  apparently  double.  From  the  distance  apart 
of  the  lines,  the  relative  velocity  of  the  stars  can  be  found,  and 
from  this  the  size  of  the  orbit  and  the  mass  of  the  stars.  Thus  it 
appears  that  in  the  case  of  Mizar  the  relative  velocity  of  the  two 
components  is  about  100  miles  per  second,  and  the  period  about  104 
days.  If  we  assume  that  the  two  stars  are  of  about  the  same  size 
(which  is  likely  since  their  spectra  are  equally  bright)  and  that  the 
orbit  is  circular,  we  find  that  the  distance  between  them  is  about 
140  000000  miles,  and  their  united  mass  about  forty  times  that 
of  the  sun. 

Vogel,  from  later  observations  (1900-1901)  shows  that  the  period 
is  only  20.6  days  (\  of  104),  which  makes  the  distance  28330000 
miles,  and  the  mass  about  9  times  that  of  the  sun. 

The  lines  in  the  spectrum  of  Beta  Aurigse  exhibit  the  same  pecu- 
liarity, but  the  doubling  occurs  once  in  two  days;  the  periodic  time 
being  four  days,  the  velocity  about  150  miles  a  second,  and  the 
diameter  of  the  orbit  about  8  000000  miles,  the  united  mass  of  the 
two  stars  comes  out  about  two  and  a  half  times  that  of  the  sun. 
These  observations  of  Professor  Pickering's  were  made  by  photo- 
graphing the  spectrum  with  the  slitless  spectroscope  (Art.  861),  and 
are  possible  only  where  the  stars  whichT  compose  the  binary  are  both 
of  them  reasonably  bright. 

In  1896  two  other  similar  cases  were  announced  by  Professor  Pickering, 
discovered  on  the  South  American  spectrum  photographs.  The  first  of  the 
two  stars  is  /A'  Scorpii,  a  star  of  the  3d  magnitude  in  thereptile's  tail.  The 
period  is  34h  42m.5,  the  relative  velocity  300  miles  a  second,  and  the  radius 
of  the  relative  orbit  6  055000  miles.  The  other  star  is  3105  Lacaille,  of  the 
4£  magnitude  in  the  constellation  Puppis ;  the  period  is  74h  46m,  with  a 
velocity  of  385  miles  a  second,  the  radius  of  the  orbit,  being  16500000 
miles.  In  both  cases  the  two  components  of  the  binary  differ  considerably 
in  brightness.  The  first-magnitude  star  Capella  is  also  a  spectroscopic 
binary  in  which  lines  of  both  components  appear. 

879*.  In  1889  Vogel  also,  as  already  stated  in  Art.  851,  detected 
with  his  spectrograph  the  orbital  motion  of  the  Algol  system,  and 
a  few  months  later  he  discovered  similar  behavior  in  the  case  of 
Spica  (a  Virginis).  At  first  the  spectroscopic  measures  upon  this 
star  appeared  very  discordant,  but  he  soon  found  that  everything 
was  reconciled  by  assuming  the  existence  of  a  companion,  too  faint 
to  be  seen,  but  massive  enough  to  make  Spica  itself  swing  around 


550  STARS,    PLANETS. 

their  common  centre  of  gravity  once  in  4d  Oh  19m,  with  a  velocity  of 
about  57  miles  a  second,  and  in  an  orbit  which,  if  circular,  has  a 
diameter  of  about  6  million  miles.  This  orbit  cannot  be  quite  edge- 
wise to  the  earth,  since  if  it  were  there  would  be  "  eclipses  "  and 
variations  in  the  brightness  of  Spica,  such  as  do  not  occur. 

Very  recently  (1895-96)  Belopolsky  has  found  by  the  same  method 
that  the  brighter  component  of  the  double  star  Castor  has,  like  Spica, 
an  invisible  companion,  which  causes  it  to  swing  backwards  and  forwards 
once  in  a  little  less  than  three  days,  with  an  orbital  velocity  of  about  15.5 
miles  a  second.  He  has  also  ascertained  that  the  short-period  variable 
8  Cephei  gives  spectroscopic  evidence  of  orbital  motion  distinctly  elliptical, 
with  a  mean  velocity  of  about  13  miles,  and  corresponding  with  the  period 
of  variation  (9d  8h.8).  Sir  Norman  Lockyer  announces  the  same  thing  with 
respect  to  17  Aquilae,  £  Geminorum,  T  Vulpeculae  and  S  Sagittse,  though  the 
details  are  not  yet  at  hand.  In  all  these  cases  the  evidence  lies  in  the  shift 
of  lines  and  not  their  doubling;  the  star  which  causes  the  observed  motion 
is  dark.  And  it  is  to  be  noted  that  in  several  of  the  cases  the  minima  and 
maxima  of  the  star  do  not  occur  where  they  ought  to  if  it  were  a  case  of 
simple  eclipse. 

18  Lyras  is  also  abundantly  proved  by  the  observations  of  Pickering, 
Lockyer,  and  several  others  to  be  a  spectroscopic  binary,  with  two,  or  per- 
haps more  than  two,  bright  components.  The  lines  of  its  spectrum  shift 
and  double,  and  behave  in  a  very  complicated  and  interesting  way.  The 
main  variations  in  its  light  are  due  to  partial  eclipses ;  but  other  causes  are 
certainly  involved.  The  dissimilarity  of  the  component  spectra  which  make 
up  the  observed  spectrum  of  this  star  leads  Pickering  to  suggest  that  we 
may  be  able  to  infer  that  certain  other  stars  with  anomalous  spectra,  com- 
bining the  characteristics  of  two  or  more  recognized  classes,  are  really  double, 
even  if  the  spectroscope  does  not  show  motion. 

In  this  connection  the  two  variable  stars  Y  Cygni  and  Z  (not  £)  Herculis 
should  be  mentioned.  From  the  peculiarities  of  their  light-curves  Dune"r 
has  shown  that  they  are  close  binaries  with  distinctly  elliptical  orbits,  having 
periods  of  about  ld  12h,  and  4d  respectively.  They  obviously  belong  to  the 
same  class  as  the  "  spectroscopic  binaries,"  though  as  yet  their  spectra  have 
not  been  satisfactorily  investigated,  owing  to  their  faintness. 

If  See's  "  tidal-evolution  "  theory  is  correct,  these  close  binaries  are  to  be 
regarded  as  infant  systems,  which  in  time  will  grow  and  widen  out.  But 
the  connection  of  bodies  nearly  equal  in  mass,  though  differing  greatly  in 
their  brightness,  is  still  a  puzzling  mystery. 

880.  Have  the  Stars  Planets  attending  them? — It  is  a  very 
natural  supposition  that  the  minute  companions  which  attend  some 
of  the  larger  stars  may  be  really  planetary  in  their  nature,  shining 


TRIPLE   AND   MULTIPLE   STARS.  551 

more  or  less  by  reflected  light.  As  to  this  we  can  only  say,  that 
while  it  is  quite  possible  that  other  stars  besides  our  sun  may 
have  their  retinues  of  planets,  it  is  quite  certain  that  such  planets 
could  not  be  seen  by  us  with  any  existing  telescope.  If  our  sun 
were  viewed  from  a  Centauri,  Jupiter  would  be  a  star  of  less  than 
the  twenty-first  magnitude,  at  a  distance  of  only  5"  from  the  sun, 
which  itself  would  be  a  smallish  first-magnitude  star. 

881.  The  statement  can  be  verified  as  follows  :  Jupiter  at  opposition  is 
certainly  not  equivalent  in  brightness  to  twenty  stars  like  Vega  (most 
photometric  measurements  make  it  from  eight  to  fourteen).  Assuming, 
however,  that  it  is  equal  to  twenty  Vegas,  its  light  received  by  the  earth 
would  be  about  ^^-^^^  of  the  sun's.  At  opposition  our  distance 
from  Jupiter  is  about  four  astronomical  units,  so  that  seen  from  the  same 
distance  as  the  sun,  its  light  would  be  sixteen  times  that  quantity,  or 
(nearly)  ^-^^  of  the  sun's. 

Now  a  ratio  of  125  000000  between  the  light  of  two  stars  corresponds  to  a 
difference  of  20  +  magnitudes 

(log  125  000000  =  8.0969  ;  but  |£jj??  =  20.24  magnitudes.  (Art.  820.)") 
y  0.4000  j 

Accordingly,  if  the  observer  were  removed  to  such  a  distance  that  the  sun 
would  appear  like  a  first-magnitude  star  (as  would  be  the  case  from 
a  Centauri),  Jupiter  would  be  a  star  of  the  twenty-first  magnitude.  Accord- 
ing to  Art.  822,  it  would  require  a  25-inch  telescope  to  show  a  star  of  the 
sixteenth  magnitude ;  it  would  therefore  require  an  instrument  with  an 
aperture  of  250  inches,  or  nearly  21  feet,  to  show  a  star  five  magnitudes 
fainter,  even  if  there  were  no  large  star  near  to  add  to  the  difficulty. 

883.  Triple  and  Multiple  Stars.  —  There  are  a  considerable  number 
of  objects  of  this  kind,  and  some  of  them  constitute  physical  systems.  In 
the  case  of  £  Cancri  the  two  larger  stars  revolve  around  their  common 
centre  in  a  nearly  circular  orbit  less  than  2"  in  diameter,  and  with  a  period 
of  about  sixty  years ;  while  the  third  star,  smaller  and  more  distant,  moves 
around  the  closed  pair  in  an  orbit  not  yet  well  determined,  but  with  a  period 
that  must  be  several  hundred  years ;  and  in  its  motion  there  is  evidence  of 
a  peculiar  perturbation,  which  Seeliger  has  satisfactorily  explained  as  the 
result  of  motion  around  an  invisible  star,  in  an  orbit  of  about  the  same  size 
as  that  of  the  principal  pair.  Dr.  See  considers  that  he  has  discovered  a 
similar  perturbation  in  the  system  of  70  Ophiuchi  due  to  an  invisible  body  ; 
but  his  conclusion  is  not  everywhere  accepted  as  yet.  In  c  Lyrae  we  have 
two  pairs,  each  making  a  very  slow  revolution,  of  periods  not  yet  determined, 
but  probably  ranging  from  300  to  500  years.  And  since  the  pairs  have  also 
a  common  proper  motion  it  is  practically  certain  that  they  also  are  physic- 
ally connected,  and  revolve  around  their  common  centre  of  gravity  in  a 


552  STAR-CLUSTERS. 

period  to  be  reckoned  by  millenniums  —  the  motion  during  the  last  hundred 
years  being  barely  perceptible.  In  other  cases,  as,  for  instance,  in  the 
multiple  star  Q  Orionis,  we  have  a  number  of  stars  not  organized  in  pairs, 
but  at  more  or  less  equal  distances  from  each  other :  we  are  confronted  by 
the  problem  of  n  bodies  in  its  most  general  and  unmanageable  form. 

STAR-CLUSTEKS. 

883.  Clusters,  —  There  are  in  the  sky  numerous  groups  of  stars 
containing  from  one  hundred  to  many  thousand  members.     Some  of 
them  are  made  up  of  stars  visible  separately  to  the  naked  eye,  as  the 
Pleiades;  some  of  them  require  a  small  telescope  to  resolve  them,  as, 
for  instance,  the  Prsesepe  in  Cancer,  and  the  group  of  stars  in  the 
sword-handle  of  Perseus ;  while  others  yet,  even  in  telescopes  of  some 
size,  look  simply  like  wisps  or  balls  of  shining  cloud,  and  break  up 
into  stars  only  in  the  most  powerful  instruments. 

In  a  large  instrument  some  of  the  telescopic  clusters  are  magnificent  ob- 
jects, composed  of  thousands  of  stellar  sparks  compressed  into  a  ball  which 
is  dazzlingly  bright  at  the  centre  and  thinning  out  towards  the  edge.  In 
some  of  them  vividly  colored  stars  add  to  the  beauty  of  the  group  and  some 
are  full  of  variable  stars  (Art.  854*).  In  the  northern  hemisphere  the  finest 
cluster  is  that  known  as  Messier  13  Herculis  (a,  16h  37m  and  S,  36°  40')  not 
very  far  from  the  "  apex  of  the  sun's  way." 

884.  The  Pleiades.  —  Of  the  naked-eye  clusters  the  Pleiades  is 
the  most  interesting  and  important.     To  an  ordinary  eye  six  stars 
are  easily  visible  in  it,  the  six  largest  ones  indicated  in  the  figure 
(Fig.  229).    Eyes  a  little  better  see  easily  five  more — those  next  in 
size  in  the  figure  (the  two  stars  of  Asterope  being  seen  as  one).     A 
very  small  telescope  (a  mere  opera-glass)  increases  the  number  to 
nearly  a  hundred;  and  with  large  instruments  more  than  400  are 
catalogued  in  the  group.     A  few  of  the  stars,  apparently  in  the 
cluster,  are  really  only  accidentally  on  the  same  line  of  vision,  and 
are  distinguished  by  proper  motions  different  from  those  of  the  rest 
of  the  group;  but  the  great  majority  have  proper  motions  nearly 
the  same  in  amount  and  direction.    Their  spectra  are  all  of  the  first 
class,  but  Alcyone  and  Pleione  show  some  of  the  hydrogen  lines 
bright. 

The  distances  and  positions  of  the  principal  stars  with  respect  to  the 
central  star  Alcyone  have  been  carefully  measured  three  or  four  times 
during  the  last  fifty  years.  The  relative  motions  during  the  period  have 


THE    PLEIADES. 


553 


not  proved  large  enough  to  admit  of  satisfactory  determination,  but  it  is 
clear  that  such  motions  exist.  A  curious  and  interesting  fact  is  the  presence 
of  nebulous  matter  in  considerable  quantity.  A  portion  of  this  nebulosity 
hanging  around  Merope  (the  northeast  star  of  the  dipper-bowl  in  the  figure) 
was  discovered  many  years  ago ;  but  it  was  reserved  to  photography  to 


Asterope 

4& 

\  '<\ 
'••-  '•• 

,-'    I 

Mala 


Alcyone/' 


-Celceno 


Electro, 


Atlas 


?*,  y 

Jf  /  Merope 

si/      \         / 


FIG.  229.  —  The  Pleiades. 

detect  very  recently  other  clouds  of  nebulosity  attached  to  other  stars,  espe- 
cially to  Maia,  and  to  show  that  the  whole  space  is  covered  with  streaks  and 
streamers  of  it,  emitting  light  of  such  a  character  as  to  impress  the  photo- 
graphic plate  much  more  strongly  than  the  eye.  The  figure  shows  roughly 
the  outlines  of  some  of  the  principal  nebulous  filaments,  but  one  must  see 
the  photographs  of  Roberts  and  others  in  order  to  get  any  adequate  idea  of 
their  extent  and  beauty. 

885.     Distance  of  Star-Clusters  and  Size  of  the  Component  Stars. 

—  The  question  at  once  arises  whether  clusters,  such  as  the  one 


554  THE   NEBULAE. 

mentioned  in  Hercules,  are  composed  of  stars  each  comparable  with 
our  sun,  and  separated  by  distances  corresponding  to  the  distance 
between  the  sun  and  its  neighboring  stars,  or  whether  the  bodies 
which  compose  the  swarm  are  really  very  small,  —  mere  sparks  of 
stellar  matter  :  whether  the  distance  of  the  mass  from  us  is  about 
the  same  as  that  of  the  stars  among  which  it  seems  to  be  set,  or 
whether  it  is  far  beyond  them.  Forty  years  ago  the  accepted  view 
was  that  the  stars  composing  the  clusters  are  no  smaller  than  ordi- 
nary stars,  and  that  the  distance  of  the  star-clusters  is  immensely 
greater  than  that  of  the  isolated  stars.  There  are  many  eloquent 
passages  in  the  writings  of  that  period  based  upon  the  belief  that 
these  star-clusters  are  stellar  universes,  —  "galaxies,"  like  the  group 
of  stars  to  which  the  writers  supposed  the  sun  to  belong,  but  so 
inconceivably  remote  that  in  appearance  they  shrank  to  these  mere 
balls  of  shining  dust. 

It  is  now,  however,  quite  certain  that  the  other  view  is  correct,  — 
that  star-clusters  are  among  our  stars  and  form  part  of  our  universe. 
Large  and  small  stars  are  so  associated  in  the  same  group  in  many 
cases  as  to  leave  us  no  choice  of  belief  in  the  matter.  It  is  true 
that  as  yet  no  parallax  has  been  detected  in  any  star-cluster ;  but 
that  is  not  strange,  since  a  cluster  is  not  a  convenient  object  for 
observations  of  the  kind  necessary  to  the  detection  of  parallax. 

THE   NEBULA. 

886.  The  Nebulae.  —  There  are  also  in  the  sky  a  multitude  of 
faintly  shining  bodies,  —  shreds  and  balls  of  cloudy  stuff  that  are 
known  as  "  nebulce "  (the  word  meaning  strictly  a  "  little  cloud "). 
About  10000  of  these  objects  are  already  catalogued. 

Two  or  three  of  them  are  visible  to  the  naked  eye.  The  nebula 
in  the  girdle  of  Andromeda  is  the  brightest  of  them,  in  which,  it 
will  be  remembered,  the  temporary  star  of  1885  appeared. 

The  next  brightest  is  the  wonderful  nebula  of  Orion,  which,  in 
the  beauty  and  variety  of  its  details,  in  the  interesting  relations  of 
the  included  stars,  the  delicate  tint  of  the  filmy  light,  and  in  its 
spectroscopic  interest,  far  exceeds  the  other,  —  indeed,  all  others. 

It  is  so  difficult  to  represent  these  delicate  objects  by  any  process 
available  in  a  text-book  that  we  limit  ourselves  to  giving  two  cuts, 
one  copied  from  Mr.  Roberts'  exquisite  photograph  of  the  great 
nebula  of  Andromeda,  and  the  other  from  a  drawing  of  the  curious 
Ring-nebula  in  Lyra. 


THE   NEBULA.  555 

The  first  successful  photograph  of  a  nebula  (the  Orion  Nebula)  was  made 
by  Dr.  H.  Draper  in  1880,  and  he  was  soon  followed  by  Common.  Since 
then  great  progress  has  been  made  both  in  Europe  and  America.  Visual 
observation  and  draughtsmanship  cannot  here  at  all  compete  with  the  photo- 
graphic process,  which  continually  brings  out  features  before  unrecognized 


FlG.  230.  —  Mr.  Roberts'  Photograph  of  the  Nebula  of  Andromeda. 

in  the  most  powerful  telescopes,  —  sometimes  new  and  startling  revelations, 
like  the  concentric  rings  in  the  Andromeda  Nebula,  an  apparent  parallel  of 
Saturn  and  his  rings.  The  photograph  has  one  drawback,  however :  stars 
in  the  Nebula  are  not  properly  shown  ;  nor  is  the  relative  brightness  of 
different  portions  fairly  given  on  any  single  negative.  The  exposure  neces- 
sary to  bring  out  faint  details  is  far  too  great  for  the  brighter  parts. 

With  a  small  telescope  a  nebula  cannot  be  distinguished  from  a 
close  star-cluster,  and  it  is  quite  likely  that  the  clusters  and  nebulaB 
shade  into  each,  other  by  insensible  gradations.  Forty  years  ago  it 
was  supposed  that  there  was  no  distinction  between  them  except 
that  of  mere  remoteness,  —  that  all  nebulae  could  be  resolved  into 
stars  by  sufficient  increase  of  telescopic  power.  When  Lord  Rosse's 


556  FORMS    AND   MAGNITUDES    OF   NEBULAE. 

great  telescope  was  first  erected,  it  was  for  a  time  reported  (and  the 
statement  is  still  often  met  with)  that  it  had  "  resolved  "  the  Orion 
nebula.  This  was  a  mistake,  however.  No  telescope  ever  has 
resolved  that  nebula  into  stars  or  ever  will,  for  we  now  know  that 
it  is  not  composed  of  stars. 

887,  Forms  and  Magnitudes  of  Nebulae. — The  larger  and  brighter 
nebulae  are,  many  of  them,  very  irregular  in  form,  stretching  out 
sprays  and  streamers  in  all  directions,  and  containing  dark  openings 
or  "  lanes."    The  so-called  "  fish-mouth  "  in  the  nebula  of  Orion,  and 
the  dark  streaks  in  the  nebula  of  Andromeda,  are  striking  examples. 
Some  of  these  bodies  are  of  enormous  volume.    The  nebula  of  Orion, 
with  its  outlying  streamers,  extends  over  several  square  degrees,  and 
the  nebula  of   Andromeda  covers  more  than  one.     Now,  as  seen 
from  even  the  nearest  star,  the  apparent  distance  of  Neptune  from 
the  sun  is  only  30",  and  the  diameter  of  its  orbit  V.     It  is  perfectly 
certain  that  neither  of  these  nebulae  is  as  near  as  a  Centauri,  and 
therefore  the  cross-section  of  the  Orion  nebula,  as  seen  from  the 
earth,  must  be  at  least  many  thousand  times  the  area  of  Neptune's 
orbit,  and  the  "  hole  "  in  the  Annular  Nebula  as  shown  in  Fig.  231 
must  be  somewhat  larger  than  that  orbit,  which  at  the  distance  of 
a  Centauri  would  subtend  an  angle  of  only  45"  on  the  scale  given  in 
the  figure. 

And  the  nebulae  as  seen  with  the  telescope  are  only  the  brightest 
portions  of  vaster  clouds. 

Recent  photographs  of  Orion,  made  with  instruments  of  short 
focus  and  with  a  long  exposure,  show  that  the  whole  constellation  is 
enveloped  in  a  nebulosity,  which  for  the  most  part  attaches  itself  to 
the  principal  stars,  like  the  nebulosity  in  the  Pleiades  (Art.  884). 
The  well-known  nebula  of  Orion  is  only  the  brightest  portion  of  this 
inconceivably  enormous  mass. 

We  do  not  know  what  is  the  real  shape  of  either  of  the  nebulae, 
whether  it  is  a  thin,  flat  sheet,  or  a  voluminous  bulk ;  but  some 
things  about  these  two  nebulae  and  several  others  favor  strongly 
the  idea  that  their  thickness  does  not  correspond  to  their  apparent 
area. 

888.  The  Smaller  Nebulae, — The  smaller  nebulae  are  for  the  most 
part  elliptical  in  outline,  some  nearly  circular,  others  more  elongated, 
and  some  narrow,  slender  streaks  of  light.      Generally  they  are 


SPECTRUM    OF    NEBULuE. 


557 


FIG.  231.  —  The  Annular  in  Lyra. 


brighter  at  the  centre,  and  in  many  cases  the  centre  is  occupied 
by  a  star.  Indeed,  there  is  a  considerable  number  of  so-called 
" nebulous  stars"  that  is,  stars 
with  a  hazy  envelope  around 
them. 

There  are  some  nebulae  which 
present  nearly  a  uniform  disc  of 
light,  and  are  known  as  "plan- 
etary" nebulae,  and  there  are 
some  which  are  dark  in  the 
centre  and  are  known  as  "an- 
nular" or  ring  nebulae.  The 
finest  of  these  annular  nebulae 
is  the  one  in  the  constellation  of 
Lyra,  about  half-way  between 
the  stars  /?  and  y;  it  is  shown 
in  Fig.  231. 

There  are  also  a  number 
of  double  nebulae,  perhaps  dou- 
ble stars  in  process  of  making. 
In  a  majority  of  the  nebulae  the  photographs  reveal  a  remarkable 
spiral  structure  of  which  the  so-called  "  whirlpool "  nebula  in 
the  constellation  "  Canes  Venatici "  is  the  most  striking  specimen. 
This  spiral  structure,  however,  is  to  be  seen  only  in  large  telescopes ; 
in  fact,  very  little  of  the  real  beauty  of  most  of  these  objects  is 
visually  accessible  to  instruments  of  less  than  12  inches  aperture. 

889.  Variable  Nebulae, — There  are  several  nebulae  which  vary  in 
their  brightness  from  time  to  time;  one  especially,  near  e  Tauri,  at 
times  has  been  visible  with  a  small  telescope,  while  at  other  times  it 
is  entirely  invisible  even  with  large  ones.     So  far  no  regular  perio- 
dicity has  been  ascertained  in  such  cases. 

890.  Their  Spectra.  —  One  of  the  earliest  and  most  remarkable 
achievements  of  the  spectroscope  was  its  demonstration  of  the  fact 
that  the  light  of  many  of  the  nebulae  proceeds  mainly  from  luminous 
gas.     They  give  a  visual  spectrum  of  six  or  seven  bright  lines1 

1  The  wave-lengths  of  these  lines  are  the  following,  in  the  order  of  brightness: 
(1)  5007.05  ?  (2)  4959.02  ?  (3)  4861.50,  Hydrogen  (F);  (4)  4340.66,  Hydrogen  (7); 
(5)  4101.85,  Hydrogen  (ft);  (6)  5875.98,  Helium  (D3);  (7)  4472,  Helium. 


558  THE   NEBULA. 

(Fig.  232),  three  of  which  are  fairly  conspicuous.     This  most  im- 
portant and  brilliant  discovery  was  made  by  Huggins  in  1864. 

Three  of  the  lines,  F,  Hy,  and  h,  are  due  to  Hydrogen.  Two,  the  faintest 
of  all,  first  observed  by  Copeland,  are  lines  of  Helium,  so  conspicuous  in  the 
solar  chromosphere;  but  the  origin  of  the  rest  remains  unknown.  The 
brightest  line  of  the  whole  number  is  in  the  green,  x,5007,  and  was  for  a 
while  referred  to  nitrogen;  but  under  closer  examination  the  identification 
breaks  down,  though  the  student  will  find  it  still  called  "  the  nitrogen  line  " 
now  and  then.  The  unidentified  element  is  provisionally  called  "  nebulium." 

Mr.  Lockyer  identified  it  with  a  "  fluting  "  in  the  low-ternperature  spec- 
trum of  magnesium,  which  he  has  found  in  the  spectrum  of  meteorites;  but 
it  is  now  certain  that  this  also  was  an  error,  and  this  line  and  its  neighbor 
4959  still  remain  a  mystery. 


FIG.  232.  —  Visual  Spectrum  of  the  Gaseous  Nebulae. 

All  the  nebulae  which  give  a  gaseous  spectrum  at  all  present  this  same 
spectrum  entire  or  in  part.  If  the  nebula  is  faint  only  the  brightest  lines 
appear,  while  the  Hy  line  and  the  other  fainter  lines  are  seen  only  in  the 
brightest  nebulae  and  under  favorable  circumstances. 

891.  Photography  since  1886  has  proved  itself  as  effective  in 
the  study  of  nebular  spectra  as  of  stellar :  we  have  at  present  a 
list  of  over  70  lines  photographed  in  the  spectra  of  half  a  dozen 
nebulae,  55  in  that  of  the  Orion  nebula  alone,  among  them  all  the 
hydrogen  lines  clear  to  the  last  of  the  ultra-violet  series.  Some  of 
the  lines  appear  also  in  the  spectra  of  the  trapezium  stars,  showing 
that  these  stars  are  of  the  same  material  as  the  surrounding  nebula, 
only  more  condensed. 

During  the  summer  of  1890,  Keeler  at  the  Lick  Observatory 
observed  a  number  of  the  planetary  nebulae  with  a  spectroscope  of 
high  dispersive  power,  and  was  able  to  detect  and  to  measure  the 
motion  of  several  of  them  in  the  line  of  sight.  The  velocity  of 
their  motion  appears  to  be  of  the  same  order  as  that  of  the  stars, 
the  nebulas  observed  giving  results  ranging  from  zero  up  to  nearly 
forty  miles  a  second,  —  some  approaching  and  some  receding. 


CHANGES   IN   THE   NEBULA.  559 

892.  But  not  all  the  nebulae  by  any  means  give  a  gaseous  spec- 
trum :  those  which  do  so  —  about  half  the  whole  number  —  are  of  a 
more  or  less  distinct  greenish  tint,  which  is  at  once  recognizable  in 
the  telescope.    The  white  nebulae,  the  nebula  of  Andromeda  at  their 
head,  give  only  a  continuous  and  perfectly  expressionless  spectrum, 
unmarked  by  any  lines  or  bands,  either  bright  or  dark.     This  must 
not  be  interpreted  as  showing  that  these  nebulae  cannot  be  gaseous; 
for  a  gas  under  pressure  gives  just  such  a  spectrum ;  but  so  also  do 
masses  of  solid  or  liquid  when  heated  to  incandescence.    The  spectro- 
scope simply  declines  to  testify  in  this  case.    The  telescopic  evidence 
as  to  the  nature  of  the  white  nebulae  is  the  same  as  for  the  green. 
They  withstand  all  attempts  at  resolution,  none  more  firmly  than 
the  Andromeda  nebula  itself,  the  brightest  of  them  all. 

893.  Changes  in  the  Nebulae.  —  The  question  has  been  raised 
whether  some  of  the  nebulae  have  not  sensibly  changed,  even  within 
the  few  years  since  it  has  become  possible  to  observe  them  in  detail. 
It  is  quite  certain  that  in  important  respects  the  early  drawings  differ 
seriously  from  those  of  recent  observers;  but  the  appearance  of  a 
nebula  depends  so  much  upon  the  telescope  and  the  circumstances 
under  which  it  is  used,  the  features  are  so  delicate  and  indefinite, 
and  the  difficulty  of  representing  them  on  paper  is  such,  that  very 
little  reliance  can  be  placed  on  discrepancies  between  drawings, 
unless  supported  by  the  evidence  of  measures  of  some  kind. 

Thus  far,  the  best  authenticated  instance  of  such  a  change,  according  to 
Professor  Holden,  is  in  the  so-called  "  trifid  "  nebula,  in  Sagittarius.  In 
this  object  there  is  a  peculiar  three-legged  area  of  darkness  which  divides 
the  nebula  into  three  lobes.  A  bright  triple  star,  which  in  the  early  part 
of  the  century  was  described  and  figured  by  Herschel  and  other  observers 
as  in  the  middle  of  one  of  these  dark  lanes,  is  now  certainly  in  the  edge  of 
the  nebula  itself.  The  star  does  not  seem  to  have  moved  with  reference  to 
the  neighboring  stars,  and  it  seems  therefore  necessary  to  suppose  that  the 
nebula  itself  has  drifted  and  changed  its  form. 

As  to  the  nebula  of  Orion,  Professor  Holden's  conclusion  is,  that  while 
the  outlines  of  the  different  features  have  probably  undergone  but  little 
change,  their  relative  brightness  and  prominence  have  been  continually  fluc- 
tuating. This,  however,  can  hardly  be  considered  certain;  to  settle  the 
question  will  probably  require  another  fifty  years  or  so,  and  the  comparison, 
not  of  drawings,  but  of  photographs. 

894.  Nature  of  the  Nebulae.  —  As  to  the  constitution  of  these 
clouds  we  can  only  speculate.    In  the  green  nebulae  we  can  say  with 


560  NUMBER   AND   DISTRIBUTION    OF    NEBULAE. 

confidence  that  hydrogen,  helium,  and  some  other  gas  are  certainly 
present,  and  that  the  gases  emit  most  of  the  light  that  reaches  us 
from  such  objects.  But  how  much  solid  or  liquid  matter  in  the  form 
of  grains  and  drops  may  be  included  within  the  gaseous  cloud  we 
have  no  means  of  knowing. 

The  idea  of  Mr.  Lockyer  (a  part  of  his  wide  induction  as  to  what  we 
may  call  the  "  meteoritic  constitution  of  the  universe  "  )  is  that  they  are 
clouds  of  "  sparse  meteorites,  the  collisions  of  which  bring  about  a  rise  of 
temperature  sufficient  to  render  luminous  one  of  their  chief  constituents," 
which,  when  he  wrote  the  sentence,  he  imagined  to  be  magnesium. 

How  far  this  theory  will  stand  the  test  of  time  and  future  investigations 
remains  to  be  seen.  At  first  view  it  seems  very  doubtful  whether  the  colli- 
sions in  such  a  body  could  be  frequent  or  violent  enough  to  account  for 
its  luminosity,  and  one  is  tempted  to  look  to  other  causes  for  the  source 
of  light.  "Luminescence  "  does  not  require  a  high  temperature. 

895.  Number  and  Distribution  of  Nebulae.  —  Sir  William  Her- 
schel  was  the  first  extensive  investigator  of  these  interesting  objects, 
and  left  his  unfinished  work  as  a  legacy  to  his  son,  Sir  John  Her- 
schel,  who  completed  the  survey  of  the  heavens  by  a  residence  of 
several  years  at  the  Cape  of  Good  Hope.  His  "General  Catalogue  " 
has  now  been  superseded  by  Dreyer's,  which  contains  about  10000 
of  them.  Photographs  have  revealed  immense  numbers  invisible  to 
the  eye  with  any  telescope.  Keeler  estimated  that  there  are  at 
least  120000  nebulae  within  reach  of  the  Crossley  reflector. 

As  to  their  distribution,  it  is  a  curious  and  important  fact  that  it 
is  in  contrast  to  the  distribution  of  the  stars.  The  stars,  as  we  shall 
soon  see,  gather  especially  in  and  about  the  Milky  Way,  as  do  also 
the  star-clusters;  but  the  nebulze  specially  crowd  together  in  regions 
as  far  from  the  Milky  Way  as  it  is  possible  to  get.  As  has  been 
pointed  out  by  more  than  one,  this  shows,  however,  not  a  want  of 
relation  between  the  stars  and  the  nebulas,  but  some  " relation  of 
contrariety."  Precisely  what  this  is,  and  why  the  nebulae  avoid  the 
regions  thickly  starred,  is  not  yet  clear.  Possibly  the  stars  devour 
them,  that  is,  gather  in  and  appropriate  surrounding  nebulosity  so 
that  it  disappears  from  their  neighborhood. 

896.  Distance  of  the  Nebulae.  —  On  this  point  we  have  very  little 
absolute  knowledge.  Attempts  have  been  made  to  measure  the  par- 
allax of  one  or  two,  but  so  far  unsuccessfully.  Still  it  is  probable, 
indeed  almost  certain,  that  they  are  at  the  same  order  of  distance  as 


THE    SIDEREAL    SYSTEM  561 

the  stars.  The  wisps  of  nebulosity  which  photography  shows  at- 
tached to  the  stars  in  the  Pleiades  (and  a  number  of  similar  cases 
appear  elsewhere),  the  nebulous  stars  of  Herschel,  and  numerous 
nebulae  which  have  a  star  exactly  in  the  centre, — these  compel  us  to 
believe  that  in  such  cases  the  nebulosity  is  really  cut  the  star.  Then 
in  the  southern  hemisphere  there  are  two  remarkable  luminous 
clouds  which  look  like  detached  portions  of  the  Milky  Way  (though 
they  are  not  near  it),  and  are  known  as  the  Nubeculse  or  "  Magellanic 
clouds."  These  are  made  up  of  stars  and  star-clusters,  and  of  nebulae 
also,  all  swarming  together,  and  so  associated  that  it  is  not  possible 
to  question  their  real  proximity  to  each  other. 

897,  Fifty  years  ago  a  very  different  view  prevailed.     As  has  been  said 
already,  astronomers  at  that  time  very  generally  believed  that  there  was  no 
distinction  between  nebulae  and  star-clusters  except  in  regard  to  distance, 
the  nebulae  being  only  clusters  too  remote  to  show  the  separate  stars.     They 
considered  a  nebula,  therefore,  as  a  "  universe  of  stars,"  like  our  own  "  galac- 
tic cluster  "  to  which  the  sun  belongs,  but  as  far  beyond  the  "  star-clusters  " 
as  these  were  believed  to  be  beyond  the  isolated  stars.     In  some  respects 
this  old  belief  strikes  one  as  grander  than  the  truth  even.     It  made  our 
vision  penetrate  more  deeply  into  space  than  we  now  dare  think  it  can. 

THE   SIDEREAL   SYSTEM. 

898,  The  Galaxy,  or  Milky  Way.— This  is  a  luminous  belt  which 
surrounds  the  heavens  nearly  in  a  great  circle.     It  varies  much  in 
width  and  brightness,  and  for  about  a  third  of   its  extent,  from 
Cygnus  to  Scorpio,  is  divided  into  two  nearly  parallel  streams.     In 
several  constellations,  as  in  Cygnus,  Sagittarius,  and  Argo  Navis, 
it  is  crossed  by  dark   straight-edged   bars  that   look   as   if   some 
light  cloud  lay  athwart  it,  and  in  the  constellation  of  Centaurus 
there  is  a  dark  pear-shaped  orifice,  —  the  "  coal  sack,"  as  it  is  called. 

The  galaxy  intersects  the  ecliptic  at  two  opposite  points  near  the 
solstices,  making  with  it  an  angle  of  about  60°.  The  northern 
"  galactic  pole,"  as  it  is  called,  lies,  according  to  Sir  John  Herschel, 
in  declination  +  27°,  and  right  ascension  12h  47m  ;  the  southern 
"  galactic  pole  "  is  of  course  at  the  opposite  point  in  the  southern 
hemisphere.  As  Herschel  remarks,  the  "  galactic  plane "  "  is  to 
sidereal  what  the  ecliptic  is  to  planetary  astronomy,  a  plane  of 
ultimate  reference,  the  ground  plan  of  the  sidereal  system." 

The  Milky  Way  is  made  up  almost  wholly  of  small  stars  from  the 
eighth  magnitude  down.  It  contains  also  a  large  number  of  star- 


562  THE   GALAXY. 

clusters,  but  (as  has  been  already  mentioned)  very  few  true  nebulae. 
In  some  places  the  stars  are  too  thickly  packed  for  counting,  es- 
pecially in  the  bright  knots  which  abound  here  and  there. 

(An  excellent  detailed  description  of  its  appearance  and  course  may  be 
found  in  Herschel's  "  Outlines  of  Astronomy." 

899.  Distribution   of   Stars   in  the   Sky :    Star-Gauges.  —  It  is 

obvious  that  the  stars  are  not  uniformly  scattered  over  the  heavens. 
They  show  a  decided  tendency  to  collect  in  groups  here  and  there, 
and  to  form  connected  streams  ;  but  besides  this,  an  enumeration  of 
the  stars  in  the  great  star-catalogues  shows  that  the  number  increases 
with  considerable  regularity  from  the  galactic  poles,  where  they  are 
most  sparse,  towards  the  galactic  circle,  where  they  are  most  crowded 
The    "star-gauges"   of  the   Herschels   make  this   fact  still  more 
obvious. 

These  gauges  consisted  merely  in  the  counting  of  the  number  of  stars 
visible  in  the  field  of  view  (15'  in  diameter)  of  the  twenty-foot  reflector. 
Sir  William  Herschel  made  3400  of  these  gauges,  directing  the  telescope  to 
different  parts  of  the  sky ;  and  his  son  followed  up  the  work  at  the  Cape  of 
Good  Hope.     Struve's  discussion  of  these  gauges  in  their  relation  to  the 
galactic  circle  gives  the  following  result :  — 

Distance  from  Galaxy.  Number  of  Stars  in  Field. 

90°          .      ...          .  .       .          .          .          .          .  4.15 

75°      ....        .        .        .     x  .        .  4.68 

60°        v .  6.52 

45° 10.36 

30° 17.68 

15°         ...         o         ....          30.30 
0°  122.00 

900.  Structure  of  the  Heavens.  —  Our  space  does  not  permit  a 
discussion  of  the  untenable  conclusions  reached  by  Herschel  and 
others  by  combining  the  unquestionable  data  derived  from  observa- 
tion, with  the  unfounded  and  untrue  assumptions  that  the  stars  are 
substantially  of  a  size  and  spaced  at  approximately  equal  distances. 
Many  of  those  conclusions  relating  to  the  form  and  dimensions  of 
the  Milky  Way,  and  of  the  stellar  universe  to  which  our  sun  belongs, 
have  become  almost  classical ;  but  they  are  none  the  less  incorrect. 

It  is  certain,  however,  that  the  faint  stars  as  a  class  are  smaller 
and  darker  and  more  remote  than  are  the  bright  ones  as  a  class:  and 
accepting  this,  we  can  safely  draw  from  the  star-gauges  a  few  general 
conclusions,  as  follows  :  — 


STRUCTURE   OF   THE   HEAVENS.  563 

We  present  them  substantially  as  given  by  Newcomb  in  his  "  Popular 
Astronomy,"  p.  491. 

1.  "  The  great  mass  of  the  stars  which  compose  this  ( stellar) 
system  are  spread  out  on  all  sides  in  or  near  a  widely  extended 
plane,  passing  through  the  Milky  Way.     In  other  words,  the  large 
majority  of  the  stars  which  we  can  see  with  the  telescope  are  con- 
tained in  a  space  having  the  form  of  a  round,  flat  disc,  the  diameter 
of  which  is  eight  or  ten  times  its  thickness. 

2.  "Within  this  space  the  stars  are  not  scattered  uniformly, but 
are  for  the  most  part  collected  into  irregular  clusters  or  masses,  with 
comparatively  vacant  spaces  between  them.'7    They  are  "  gregarious," 
to  use  Miss  Clerke's  expression. 

3.  Our  sun  is  near  the  centre  of  this  disc-like  space. 

4.  The  naked-eye  stars  "  are  scattered  in  this  space  with  a  near 
approach  to  uniformity,"  the  exceptions  being  a  few  star-clusters  and 
star-groups  like  the  Pleiades  and  Coma  Berenices. 

5.  "The  disc  described  above  does  not  represent  the  form  of  the 
stellar  system,  but  only  the  limits  within  whicli  it  is  mostly  con- 
tained."    The  circumstances  are  such  as^  "  prevent  our  assigning 
any  more  definite  form  to  the  system  than  we  could  assign  to  a  cloud 
of  dust." 

6.  "  On  each  side  of  the  galactic  region  the  stars  are  more  evenly 
and  thinly  scattered,  but  probably  do  not  extend  out  to  a  distance  at 
all  approaching  the  extent  of  the  galactic  region,"  or  if  they  do  they 
are  very  few  in  number  ;  but  it  is  impossible  to  set  any  definite 
boundaries. 

7.  On  each  side  of  the  galactic  and  stellar  region  we  have  a 
nebular  region,  comparatively  starless,  but  occupied  by  great  num- 
bers of  nebulae. 

As  to  the  Milky  Way  itself,  it  is  not  yet  certain  whether  the  stars 
which  compose  it  are  distributed  pretty  equally  near  the  galactic 
circle,  or  whether  they  form  something  like  a  ring  with  a  compara- 
tively vacant  space  in  the  middle. 

As  to  the  distance  of  the  remotest  stars  in  the  stellar  system,  it  is 
impossible  to  say  .anything  very  definite,  but  it  seems  quite  certain 
that  it  must  be  at  least  as  great  as  10000  to  20000  light-years.  If 
one  asks  what  is  beyond,  whether  the  star-filled  space  extends  in- 
definitely or  not,  no  certain  answer  can  be  given. 

Nor  is  there  now  any  reason  to  suppose  that  our  own  stellar  system  is  sepa- 
rated from  other  stellar  systems  by  any  vast  abyss  of  practically  empty 


564  STRUCTURE    OF   THE   HEAVENS. 

space,  relatively  proportioned  to  that  which  separates  our  planetary  system 
from  the  possible  planetary  systems  of  other  suns. 

901.  Do  the  Stars  Form  a  System?  —  That  is,  do  they  form  an 
organized  unit,  in  which,  as  in  the  solar  system,  each  of  the  different 
members  has  its  own  function  and  permanently  maintains  its  relation 
to  the  rest  ?     Gravitation  probably  operates,  as  indicated  *  by  the 
binary  stars,  and  the  stars  are  moving  swiftly  in  various  directions 
with  enormous  velocities,  as  shown  by  their  proper  motions,  and  by 
the  spectroscope.     The  question  is  whether  these  motions  are  con- 
trolled by  gravitation,  and  whether  they  carry  the  stars  in  orbits  that 
can  be  known  and  predicted. 

That  the  stars  are  organized  into  a  system  or  systems  of  some  sort 
can  hardly  be  doubted.  But  that  the  system  is  one  at  all  after  the 
pattern  of  the  solar  system,  in  which  the  different  members  move 
in  closed  orbits,  —  orbits  that  are  permanent  except  for  the  slow 
changes  produced  by  perturbation,  —  this  is  almost  certainly  im- 
possible, as  was  said  a  few  pages  back. 

902.  Is  there  a  Revolution  of  the  Whole  Mass  of  Stars?  — A 

favorite  idea  has  been  that  the  mass  of  stars  which  constitutes  our  system 
has  a  slow  rotation  like  that  of  a  body  on  its  axis,  the  plane  of  this  general 
revolution  coinciding  with  the  plane  of  the  galaxy.  Such  a  general  motion 
is  not  in  any  way  inconsistent  with  the  independent  motions  of  the  indi- 
vidual stars,  and  there  is  perhaps  a  slight  inherent  probability  in  favor  of 
such  a  movement ;  but  thus  far  we  have  no  evidence  that  it  really  exists — 
indeed,  there  hardly  could  be  any  such  evidence  at  present,  because  exact 
Astronomy  is  not  yet  old  enough  to  have  gathered  the  necessary  data. 

903.  Central  Snns.  —  A  number  of  speculative  astronomers,  Madler 
perhaps  most  prominently,  have  held  the  belief  that  there  is  a  "central  sun" 
standing  in  some  such  relation  to  the  stellar  system  as  our  sun  does  to  the 

JThey  fall  short  of  "demonstrating'1'1  it,  because,  although  their  apparent 
motions  are  perfectly  consistent  with  the  universality  of  gravitation,  they  are 
equally  so  with  several  other  imaginable  laws  of  force.  (See  Article  by  A.  Hall, 
Astron.  Journal,  Vol.  VIII.)  But  all  other  laws  involve  the  improbable  condi- 
tion that  the  force  must  vary  with  the  direction  as  well  as  the  distance.  Spectro- 
scopic  observations  of  the  velocity  of  binaries  are,  however,  theoretically  com- 
petent to  decide  the  question  by  enabling  us  to  compute  the  actual  orbit  of  a 
binary  from  its  apparent  orbit  (as  given  by  the  micrometer)  without  any  assump- 
tions as  to  the  law  of  the  central  force.  (See  Art.  873.)  Dr.  See  has  recently 
worked  out  the  necessary  formulae  very  completely. 


STRUCTURE    OF    THE   HEAVENS.  565 

solar  system.  It  is  hardly  necessary  to  say  that  the  notion  has  not  the 
slightest  foundation,  or  even  probability. 

Lambert  supposed  many  such  suns  as  the  centres  of  subordinate  stellar 
systems,  and  because  we  cannot  see  them,  he  imagined  them  to  be  dark. 

If  we  conceive  of  boundaries  drawn  around  our  stellar  system,  and  count 
all  the  stars  within  the  limits  as  members  of  it,  leaving  out  of  the  account 
all  that  fall  outside,  then,  of  course,  our  system  so  limited  has  at  any  moment 
a  perfectly  definite  centre  of  gravity.  There  is  no  reason  why  some  particular 
star  may  not  be  very  near  that  centre,  and  in  that  sense  a  "  central  sun  "  is 
possible  ;  but  its  central  position  would  not  give  it  any  preeminence  or  rule 
over  its  neighbors,  or  put  it  in  any  such  relation  to  the  rest  of  the  stars  as 
the  sun  bears  to  the  planets. 

904,  Orbits  of  Sun  and  Stars.  —  It  is  practically  certain  that 
the  motions  of  the  stars  are  not  orbital  in  any  strict  sense.  Except- 
ing stars  which  are  in  clusters,  all  other  stars  are  simultaneously 
acted  upon  by  many  forces  drawing  in  various  and  opposite  direc- 
tions ;  and  these  forces  must  in  most  cases  be  so  nearly  balanced 
that  the  resultant  cannot  be  very  large.  The  motions  of  the  stars 
must  consequently,  as  a  rule,  be  nearly  rectilinear. 

Still  the  balancing  of  the  forces  will  seldom  be  exact,  and  accord- 
ingly the  path  of  a  star  will  almost  always  be  slightly  curved  ;  and 
since  the  amount  and  direction  of  the  resultant  force  which  acts  on 
the  star  is  continually  changing,  the  curvature  of  its  motion  will 
alter  correspondingly,  and  the  result  will  be  a  path  which  does  not 
lie  in  any  one  plane,  but  is  bent  about  in  all  ways  like  a  piece  of 
crooked  wire.  It  is  hardly  likely,  however,  that  the  curvature  of  a 
star's  path  would,  in  any  ordinary  case,  be  such  as  could  be  detected 
by  the  observations  of  a  single  century,  or  even  of  a  thousand 
years. 

As  has  been  said  before,  in  connection  with  the  proper  motions 
of  the  stars,  the  probability  is  that  the  separate  stars  move  nearly 
independently,  "  like  bees  in  a  swarm."  In  the  solar  system  the 
central  power  is  supreme,  and  perturbations  or  deviations  from  the 
path  which  the  central  power  prescribes  are  small  and  transient.  In 
the  stellar  system,  on  the  other  hand,  the  central  force,  if  it  exists 
at  all  (as  an  attraction  towards  the  centre  of  gravity  of  the  whole 
mass  of  stars)  is  trifling.  Perturbation  prevails  over  regularity, 
and  "individualism"  is  the  method  of  the  greater  system  of  the 
stars,  as  solar  despotism  is  that  of  the  smaller  system  of  the 
planets. 


566  COSMOGONY. 

905.  Cosmogony.  —  Unquestionably  one  of  the  most  interesting, 
and  also  most  baffling,  topics  of  speculation  is  the  problem  of  the 
way  in  which  the  present  condition  of  the  universe  came  about.     By 
what  processes  have  moons  and  earths  and  Jupiters  and  Saturns, 
come  to  their  present  state  and  into  their  relation  to  the  sun  ?    What 
has  been  their  past  history,  and  what  has  the  future  in  store  for 
them  ?     How  has  the  sun  come  to  his  present  glory  and  dominion  ? 
and  in  the  stellar  universe,  what  is  the  meaning  and  mutual  relation 
of  the  various  orders  of  bodies  we  see,  —  of  the  nebulae,  the  star- 
clusters,  and  the  stars  themselves  ? 

In  a  forest,  to  use  a  comparison  long  ago  employed  by  the  elder 
Herschel,  we  see  around  us  trees  in  all  stages  of  their  life-history. 
There  are  the  seedlings  just  sprouting  from  the  acorn,  the  slender 
saplings,  the  sturdy  oaks  in  their  full  vigor,  those  also  that  are  old 
and  near  decay,  and  the  prostrate  trunks  of  the  dead.  Can  we 
apply  the  analogy  to  the  heavens,  and  if  we  can,  which  of  the  objects 
before  us  are  to  be  regarded  as  in  their  infancy,  and  which  of  them 
as  old  and  near  dissolution  ? 

906.  Fundamental   Principles  of  a  Rational  Cosmogony.  —  In 

the  present  state  of  science  many  of  the  questions  thus  suggested 
seem  to  be  hopelessly  beyond  the  reach  of  investigation,  while 
others  appear  like  problems  which  time  and  patient  work  will  solve, 
and  others  yet  have  already  received  clear  and  decided  answers. 
In  a  general  way  it  may  be  said  that  the  condensation  and  aggrega- 
tion of  rarefied  masses  of  matter  under  the  force  of  gravitation ;  the 
conversion  into  heat  of  the  (potential )  "  energy  of  position  "  destroyed 
by  the  process  of  condensation ;  the  effect  of  this  heat  upon  the  con- 
tracting mass  itself,  and  the  radiation  of  energy  into  space  and  to  sur- 
rounding bodies  as  waves  of  light  and  heat,  —  these  principles  contain 
nearly  all  the  explanations  that  can  thus  far  be  given  of  the  present 
state  of  the  heavenly  bodies. 

907.  The  Planetary  System.  —  We  see  that  our  planetary  system 
is  not  a  mere  accidental  aggregation  of  bodies.     Masses  of  matter 
coming  hap-hazard  towards  the  sun  would  move,  as  comets  do,  in 
orbits,  always  conic  sections  to  be  sure,  but  of  every  degree  of  eccen- 
tricity and  inclination.     There  are  a  multitude  of  relations  actually 
observed  in  the  planetary  system  which  are  wholly  independent  of 
gravitation  and  demand  an  explanation. 


THE   NEBULAR    HYPOTHESIS.  567 

1.  The  orbits  are  all  nearly  circular. 

2.  They  are  all  nearly  in  one  plane  (excepting  the  cases  of  some 
of  the  little  asteroids). 

3.  The  revolution  of  all  is  in  the  same  direction. 

4.  There  is  a  curiously  regular  progression  of  distance  (expressed 
by  Bode's  law,  which,  however,  breaks  down  at  Neptune). 

5.  There  is  a  roughly  regular  progression  of  density,  increasing 
both  ways  from  Saturn,  the  least  dense  of  all  the  planets  in  the 
system. 

As  regards  the  planets  themselves,  we  have 

6.  The  plane  of  the  planets7  rotation  nearly  coinciding  with  that  of 
the  orbit  (probably  excepting  Uranus). 

7.  The  direction  of  the  rotation  the  same  as  that  of  the  orbital 
revolution  (excepting  probably  Uranus  and  Neptune). 

8.  The  plane  of  orbital  revolution  of  the  satellites  coinciding  nearly 
with  that  of  the  planet's  rotation. 

9.  The  direction  of  the  satellites'  revolution  also  coinciding  with 
that  of  the  planet's  rotation. 

10.    The  largest  planets  rotate  most  swiftly. 

908.  Origin  of  the  Nebular  Hypothesis.  —  Now  this  is  evidently  a 
good  arrangement  for  a  planetary  system,  and  therefore  some  have  inferred 
that  the  Deity  made  it  so,  perfect  from  the  first.     But  to  one  who  considers 
the  way  in  which  other  perfect  works  of  nature  usually  come  to  their  perfec- 
tion —  their  processes  of  growth  and  development  —  this  explanation  seems 
improbable.     It  appears  far  more  likely  that  the  planetary  system  grew  than 
that  it  was  built  outright. 

Three  different  philosophers  in  the  last  century,  Swedenborg,  Kant,  and 
La  Place  (only  one  of  them  an  astronomer),  independently  proposed  essen- 
tially the  same  hypothesis  to  account  for  the  system  as  we  now  know  it. 
La  Place's  theory,  as  might  have  been  expected  from  his  mathematical  and 
scientific  attainments,  was  the  most  carefully  and  reasonably  worked  out  in 
detail.  It  was  formulated  before  the  discovery  of  the  great  principle  of  the 
"conservation  of  energy,"  and  before  the  mechanical  equivalence  of  heat 
with  other  forms  of  energy  was  known,  so  that  in  some  respects  it  is  defec- 
tive, and  even  certainly  wrong.  In  its  main  idea,  however,  that  the  solar 
system  once  existed  as  a  nebulous  mass  and  has  reached  its  present  state  as 
the  result  of  a  series  of  purely  physical  processes,  it  seems  certain  to  prove 
correct,  and  it  forms  the  foundation  of  all  the  current  speculations  upon  the 
subject. 

909.  La  Place's  Theory.  —  (a)  He  supposed  that  at  some  past 
time,  which  may  be  taken  as  the   starting-point  of  our  system's 


568  COSMOGONY. 

history  (though  it  is  not  to  be  considered  as  the  beginning  of  the 
existence  of  the  substance  of  which  our  system  is  composed),  the 
matter  now  collected  in  the  sun  and  planets  was  in  the  form  of  a 
nebula. 

(b)  This  nebula  was  a  cloud  of  intensely  heated  gas,  perhaps  hotter, 
as  he  supposed,  than  the  sun  is  now. 

(c)  This  nebula,  under  the  action  of  its  own  gravitation,  assumed 
an  approximately  globular  form  with  a  rotation  around  an  axis.    As 
to  this  movement  of  rotation,  it  appears  to  be  necessary  to  account 
for  it  by  supposing  that  the  different  portions  of  the  nebula,  before 
the  time  which  has  been  taken  as  the  starting-point,  had  motions  of 
their  own.     Then,  unless  these  motions  happened  to  be  balanced  in 
the  most  perfect  and  improbable  manner,  a  motion  of  rotation  would 
set  in  of  itself  as  the  nebula  contracted,  just  as  water  whirls  in  a 
basin  when  drawn  off  by  an  orifice  in  the  bottom.     The  velocity  of 
this  rotation  would  become  continually  swifter  as  the  volume  of  the 
nebula  diminished,  the  so-called  "moment  of  momentum'7  remaining 
necessarily  unchanged. 

910.  (d)  In  consequence  of  this  rotation,  the  mass,  instead  of 
remaining  spherical,  would  become  much  flattened  at  the  poles,  and 
as  the  rotation  went  on  and  the  motion  became  accelerated,  the  time 
would  come  when  the  centrifugal  force  at  the  equator  of  the  nebula 
would  become  equal  to  gravity,  and  "rings  of  nebulous  matter" 
would  be  abandoned  (not  thrown  off),  resembling  the  rings  of  Saturn, 
which,  indeed,  suggested  this  feature  of  the  theory. 

(e)  A  ring  would  revolve  for  a  while  as  a  whole,  but  in  time  would 
break,  and  the  material  would  collect  into  a  single  globe.  La  Place 
supposed  that  the  ring  would  revolve  as  if  it  were  solid,  the  outer 
edge,  therefore,  moving  more  swiftly  than  the  inner.  If  this  were 
so,  the  mass  formed  from  the  collection  of  the  matter  of  the  ruptured 
ring  would  necessarily  rotate  in  the  same  direction  as  the  ring  had 
revolved. 

(/)  The  planet  thus  formed  would  continue  to  revolve  around  the 
central  mass,  and  might  itself  in  turn  abandon  rings  which  might 
break,  and  so  furnish  it  with  a  retinue  of  satellites. 

911.  It  is  obvious  that  this  theory  meets  completely  most  of  the 
conditions  of  the  problem.     It  explains  every  one  of  the  facts  just 
mentioned  as  demanding  explanation  in  the  solar  system.     Indeed, 


MODIFICATIONS    OF   LA   PLACE'S   THEORY.  569 

it  explains  them  almost  too  well ;  for  as  the  theory  stands  it  meets 
a  most  serious  difficulty  in  the  exceptional  cases  of  the  planetary 
system,  such  as  the  anomalous  and  retrograde  revolutions  of  the 
satellites  of  Uranus  and  Neptune.  Another  difficulty  lies  in  the 
swift  revolution  of  Phobos  (Art.  590),  the  inner  satellite  of  Mars. 
According  to  the  unmodified  nebular  hypothesis,  no  planet  or 
satellite  could  have  a  time  of  revolution  less  than  the  time  of 
rotation  which  the  central  body  would  have,  if  expanded  until 
its  radius  becomes  equal  to  the  radius  of  the  satellite's  orbit ; 
still  less  could  it  have  a  period  shorter  than  the  central  body 
now  has. 

912.  Necessary  Modifications.  —  The  principal  modifications 
which  seem  essential  to  the  theory  in  the  light  of  our  present 
knowledge,  are  the  following.  (The  small  letters  indicate  the 
articles  of  the  original  theory  to  which  reference  is  made.) 

(b)  It  is  not  probable  that  the  original  nebula  could  have  been  at 
a  temperature  even  nearly  as  high  as  the  present  temperature  of  the 
sun.  The  process  of  condensation  of  a  gaseous  cloud  from  loss  of 
heat  by  radiation  would  cause  the  temperature  to  rise,  according  to 
the  remarkable  and  almost  paradoxical  law  of  Lane  (Art.  357), 
until  the  mass  had  begun  to  liquefy  or  solidify.  And  it  appears 
probable  that  the  original  nebula,  instead  of  being  purely  gaseous, 
was  rather  a  cloud  of  dust  than  a  "fire-mist" ';  i.e.,  that  it  was  made 
up  of  finely  divided  particles  of  solid  or  liquid  matter,  each  particle 
enveloped  in  a  mantle  of  permanent  gas.  Such  a  nebula  in  con- 
densing would  rise  in  temperature  at  first  as  if  purely  gaseous,  so 
that  its  central  mass  after  a  time  would  reach  the  solar  stage  of 
temperature,  the  solid  and  liquid  particles  melting  and  vaporizing 
as  the  mass  grew  hotter.  At  a  subsequent  stage,  when  yet  more  of 
the  original  energy  of  the  mass  had  been  dissipated  by  radiation, 
the  temperature  of  the  bodies  which  were  formed  from  and  within 
the  nebula  would  fall  again. 

And  yet  La  Place  may  have  been  right  in  ascribing  a  high  temperature  to 
the  original  nebula.  If  that  were  really  the  case,  the  only  difference  would 
be  that  the  nebula  would  be  longer  in  reaching  the  condition  of  a  solar 
system ;  but  it  is  not  necessary,  as  he  supposed,  to  assume  that  the  original 
temperature  was  high,  and  that  the  matter  was  originally  in  a  purely  gaseous 
condition,  in  order  to  account  for  the  present  existence  of  such  a  group  of 
bodies  as  the  incandescent  sun  and  its  cool  attendant  planets. 


570  COSMOGONY. 

•^ 

913.  (d)  As  regards  the  manner  in  which  the  planetary  bodies 
were  probably  liberated  from  the  parent  mass,  it  seems  to  be  very 
doubtful  whether  the  matter  accumulated  at  the  equator  of  the  rotat- 
ing mass  would  usually  separate  itself  as  a  ring.     If  a  plastic  mass 
in  swift  rotation  is  not  absolutely  homogeneous  and  symmetrical,  it 
is  more  likely  to  become  distorted  by  a  lump  formed  somewhere  on 
its  equator,  which  lump  may  be  finally  detached  and  circulate  around 
its  primary.    The  formation  of  a  ring,  though  possible,  would  seem 
likely  to  be  only  a  rare  occurrence. 

La  Place  seems  to  have  believed  also  that  the  outer  rings  must 
necessarily  have  been  abandoned  first,  and  the  others  in  regular  suc- 
cession, so  that  the  outer  planets  are  much  the  older.  It  seems,  how- 
ever, quite  possible,  and  even  probable,  that  several  of  the  planets 
may  be  of  about  the  same  age,  more  than  one  ring  having  been 
liberated  at  the  same  time ;  or  several  planets  having  been  formed 
from  different  zones  of  the  same  ring. 

(e)  In  the  case  where  a  ring  was  formed,  it  is  practically  certain 
that  it  could  not  have  revolved  as  a  solid  sheet ;  i.e.,  with  the  same 
angular  velocity  for  all  the  particles,  and  with  the  outer  portions, 
therefore,  moving  more  swiftly  than  the  inner.  If,  for  instance,  the 
matter  which  now  constitutes  the  earth  were  ever  distributed  to  form 
a  ring  occupying  anything  like  half  the  distance  from  Venus  to  Mars, 
it  must  have  been  of  a  tenuity  comparable  only  to  that  of  a  comet. 
The  separate  particles  of  such  a  ring  could  have  had  very  little  con- 
trol over  each  other,  and  must  have  moved  independently ;  the  outer 
ones,  like  remoter  planets,  making  their  circuits  in  longer  periods 
and  moving  more  slowly  than  those  near  the  inner  edge,  as  is  now 
known  to  be  the  case  with  Saturn's  rings  (Art.  641*). 

914.  Trowbridge's  Explanation  of  the  Anomalous  Rotation  of 
Uranus  and  Neptune. — When  such  a  ring  concentrates  into  a  single 
mass,  the  direction  of  the  rotation  of  the  resultant  planet  depends 
upon  the  manner  in  which  the  matter  was  originally  distributed. 
If  the  ring  be  nearly  of  the  same  density  throughout,  the  resulting 
planet  (which  would  be  formed  at  about  the  middle  of  the  ring's 
width)  must  have  a  retrograde  rotation  like  Uranus  and  Neptune. 
But  if  the  particles  of  the  ring  are  more  closely  packed  near  its 
inner  edge,  so  that  the  resultant  planet  would  be  formed  much  within 
the  middle  of  its  width,  its  axial  rotation  must  be  direct.     In  the 
first  case,  illustrated  in  Fig.  233  (a),  the  particles  near  the  inner  edge 


THE  ROTATION  OF  THE  PLANETS. 


571 


of  the  ring  would  control  the  rotation,  having  a  greater  moment  of 
rotation  with  respect  to  M,  where  the  planet  is  supposed  to  be  formed, 
than  those  at  the  outer  edge.  The  rotation,  therefore,  will  be  retro- 
grade, on  account  of  their  greater  velocity. 

In  the  other  case,  Fig.  233  (£),  where  the  inner  edge  of  the  ring 
is  densest,  and  the  planet  is  formed  as  at  JV,  much  nearer  the  inner 
than  the  outer  edge  of  the  ring,  the  aggregate  moment  of  rotation 


FIG.  233.  —  Rotation  of  Planets  formed  from  Rings  according  to  Trowbridge. 

with  respect  to  JV  is  greater  for  the  particles  beyond  N  (because  of 
their  greater  distance  from  it)  than  that  of  the  swifter  moving  parti- 
cles within,  and  this  determines  a  direct  rotation. 

The  fact  that  the  satellites  of  Uranus  and  Neptune  revolve  back- 
wards is  not,  therefore,  at  all  a  bar  to  the  acceptance  of  the  nebular 
hypothesis,  as  sometimes  represented.  If  a  new  planet  should  ever 
be  discovered  outside  of  Neptune  it  is  altogether  probable  that  its 
satellites  would  be  found  to  retrograde. 

This  is  not  the  only  way  in  which  the  retrograde  rotation  of  the  outer 
planets  may  be  accounted  for.  The  theory  of  "tidal  evolution"  (Art.  916) 
indicates  ways  in  which  an  original  rotation  might  be  reversed,  and  its  period 
greatly  changed.  See  also  the  next  Article. 

915.  Faye  in  1884  propounded  a  modification  of  the  nebular  hypoth- 
esis which  makes  the  planets  of  the  "  terrestrial  group  "  (Mercury,  Venus, 


572  COSMOGONY. 

the  Earth,  and  Mars)  older  than  the  outer  ones.1  He  supposes  that  the 
planets  were  formed  by  local  condensations  (not  by  the  formation  of  rings) 
within  the  revolving  nebula.  At  first,  before  the  nebula  was  much  condensed 
at  the  centre,  the  inward  attraction  would  be  at  any  point  directly  propor- 
tional to  the  distance  of  that  point  from  the  centre  of  gravity  of  the  nebula;  i.e., 
the  force  could  be  expressed  by  the  equation  F=  or.  After  the  condensation 
has  gone  so  far  that  practically  almost  the  whole  of  the  matter  is  collected 
at  the  centre  of  the  nebula,  the  force  is  inversely  proportional  to  the  square  oj 
the  distance,  —  the  ordinary  law  of  gravitation, 

«'•«•»  F=^' 

At  any  intermediate  time,  during  the  gradual  condensation  of  the  nebula, 
the  intensity  of  the  central  force  will,  therefore,  be  given  by  an  expression 
having  the  form 

F= 


r  being  the  distance  of  the  body  acted  upon  from  the  centre  of  gravity  of 
the  nebula,  while  a  and  b  are  coefficients  which  depend  upon  its  age  ;  a  con- 
tinually decreasing  as  the  nebula  grows  older,  while  b  increases.  The  planets 
formed  within  the  nebula  when  it  was  young,  i.e.,  when  a  was  large  and  b 
was  small,  would  have  direct  rotation  upon  their  axes,  while  those  formed 
after  a  had  sensibly  vanished  would  have  a  retrograde  rotation  ;  and  this 
he  supposes  to  be  the  case  with  Uranus  and  Neptune,  which  he  considers 
younger  than  the  inner  planets.  Faye's  work  "  L'Origine  du  Monde,"  1885, 
contains  an  excellent  summary  of  the  views  and  theories  of  the  different 
astronomers  who  have  speculated  upon  the  cosmogony. 

916.  Tidal  Evolution.  —  About  1885  Prof.  George  H.  Darwin 
(son  of  the  great  naturalist)  made  some  important  investigations 
upon  the  effect  of  tidal  reaction  between  a  central  mass  and  a  body 
revolving  about  it,  both  of  them  being  supposed  to  be  of  such  a 
nature  (i.e.,  not  absolutely  rigid)  that  tides  can  be  raised  upon 
them  by  their  mutual  attraction.  We  have  already  alluded  to  the 
subject  in  connection  with  the  tides  (Art.  484).  He  finds  in  this 
reaction  an  explanation  of  many  puzzling  facts.  It  appears,  for 
instance,  that  if  a  planet  and  its  satellite  have  ever  had  their  times 
of  rotation  of  the  same  length  as  the  time  of  their  orbital  revolution 
around  their  common  centre  of  gravity,  then,  starting  from  that 
time,  either  of  two  things  might  happen,  —  the  satellite  might  begin 
to  recede  from  the  planet,  or  it  might  fall  back  to  the  central  mass. 
The  condition  is  one  of  unstable  equilibrium,  and  the  slightest  cause 

1  If  Mars  ultimately  proves  to  be  warmer  than  the  earth  (see  Art.  589)  it  will 
be  a  strong  argument  in  favor  of  Faye's  hypothesis. 


TIDAL   EVOLUTION.  573 

might  determine  the  subsequent  course  of  things  in  either  of  the 
two  opposite  directions.  Whenever  the  time  of  rotation  of  the 
planet  is  shorter  than  the  orbital  period  of  the  satellite  (as  it  would 
naturally  become  by  condensation  continuing  after  the  separation  of 
the  satellite),  the  tendency  would  be,  as  explained  in  Art.  484, 
slightly  to  accelerate  the  satellite,  and  so  to  cause  it  continually  to 
recede  by  an  action  the  reverse  of  that  produced  by  the  hypothetical 
resisting  medium  which  is  supposed  to  disturb  Encke's  comet.  This, 
it  will  be  remembered,  is  thought  to  be  the  case  with  our  moon. 

917.  But  if  by  any  means  the  rotation  of  the  planet  were  retarded, 
so  that  its  day  should  become  longer  than  the  period  of  the  satellite, 
the  tides  produced  by  the  satellite  upon  the  planet  will  then  retard 
the  motion  of  the  satellite  like  a  resisting  medium,  and  so  will  cause 
a  continual  shortening  of  its  period,  precisely  as  in  the  case  of 
Encke's  comet.     If  nothing  intervenes,  this  action  will  in  time  bring 
down  the  satellite  upon  the  planet's  surface.     Now  in  the  case  of 
Mars  there  is  a  known  cause  operating  to  retard  its  rotation  (namely, 
the  tides  which  are  raised  by  the  sun  upon  the  planet),  and  those 
who  accept  the  theory  of  tidal  evolution  suggest  that  this  was  the 
cause  which  first  made  the  length  of  the  planet's  day  to  exceed  the 
period  of  the  satellite,  and  so  enabled  the  planet  to  establish  upon 
the  satellite  that  retardation  which  has  shortened  its  little  month, 
and  must  ultimately  bring  it  down  upon  the  planet. 

Processes  such  as  these  of  tidal  evolution  must  necessarily  be 
extremely  slow.  How  long  are  the  periods  involved,  no  one  can  yet 
estimate  with  any  precision,  but  it  is  certain  that  the  years  are  to 
be  counted  by  the  million. 

(We  have  already  referred  the  reader  (Art.  484*)  to  the  last  chapter  of 
Ball's  "  Story  of  the  Heavens  "  as  containing  an  excellent  and  easily  under- 
stood explanation  of  this  subject.) 

918.  Conclusions  derived  from  the  Theory  of  Heat.  —  As  Profes- 
sor Newcomb  has  said,  "  Kant  and  La  Place  seem  to  have  arrived 
at  the  nebular  hypothesis  by  reasoning  forwards.     Modern  science 
obtains  a  similar  result  by  reasoning  backwards  from  actions  which 
we  now  see  going  on  before  our  eyes." 

We  have  abundant  evidence  that  the  earth  was  once  at  a  much 
higher  temperature  than  now.  As  we  penetrate  below  the  surface 
we  find  the  temperature  continually  rising  at  a  rate  of  about  1°  F. 
for  every  fifty  or  sixty  feet,  thus  indicating  that  at  the  depth  of  a 


574  COSMOGONY. 

few  miles  the  temperature  must  be  far  above  incandescence.  Now, 
since  the  surface  temperature  is  so  much  lower,  this  implies  one  (or 
both)  of  two  things,  —  either  that  heat-making  processes  are  going 
on  within  the  earth  (which  may  be  true  to  some  extent),  or  else  that 
the  earth  has  been  much  hotter  than  it  now  is,  and  is  cooling  off,  — 
and  this  seems  to  be  a  most  probable  supposition.  It  is  just  as 
reasonable,  as  Lord  Kelvin  expresses  it,  to  suppose  that  the  earth 
has  lately  been  intensely  heated  as  to  suppose  that  a  warm  stone 
that  one  picks  up  in  the  field  has  been  lately  somewhere  in  the  fire. 

919.  Evidence  derived  from  the   Condition  of  the  Moon  and 
Planets.  —  In  the  case  of  the  moon  we  find  a  body  bearing  upon  its 
surface  all  the  marks  of  past  igneous  action,  but  now  in  appearance 
intensely  cold.     The  planets,  so  far  as  we  can  judge  from  what  we 
can  see  through  the  telescope,   corroborate  the  same  conclusion. 
Their  testimony  is  not  very  strong,  but  it  is  at  least  true  that 
nothing  in  the  aspect  of  any  of  them  militates  against  the  view  that 
they  also  are  bodies  cooling  like  the  earth ;  and  in  the  cases  of 
Jupiter  and  Saturn  many  phenomena  go  to  show  that  they  are  still 
(or  at  least  now)  at  a  high  temperature,  —  as  might  be  expected  of 
bodies  of  such  an  enormous  mass,  which,  necessarily,  other  things  being 
equal,  would  cool  much  more  slowly  than  smaller  globes  like  the  earth. 

The  ratio  of  surface  to  mass  is  smaller  as  the  diameter  of  a  globe  grows 
larger,  and  upon  this  ratio  the  rate  of  cooling  of  a  body  largely  depends.  In 
short,  everything  we  can  ascertain  from  the  observation  of  the  planets  agrees 
completely  with  the  idea  that  they  have  come  to  their  present  condition  by 
cooling  down  from  a  molten  or  even  gaseous  state. 

920.  The  Sun's  Testimony.  —  In  the  sun  we  have  a  body  steadily 
pouring  forth  an  absolutely  inconceivable  amount  of  heat,  without 
any  visible  source  of  supply.     Thus  far  the  only  reasonable  hypoth- 
esis to  account  for  this,  and  for  a  multitude  of  other  phenomena 
which  it  shows  us,  is  the  one  which  makes  it  a  great  cloud-mantled 
ball  of  incandescent  gases,  slowly  shrinking  under  its  own  central 
gravity,  converting  continually  a  portion  of  its  "  potential  energy  of 
position  "  *  into  the  kinetic-energy  of  heat,  which  at  present  is  mainly 
radiated  off  into  space. 

1  -By  "potential  energy  of  position  "  is  meant  the  energy  due  to  the  separated 
condition  of  its  particles  from  each  other.  As  they  fall  together  and  towards 
the  centre  in  the  shrinkage  of  the  sun,  they  "do  work"  in  precisely  the  same 
way  as  any  falling  weight. 


CONSIDERATIONS   FROM   THE   THEORY   OF    HEAT.         575 

We  say  mainly,  because  it  is  not  impossible  that  the  sun's  temperature  is 
even  yet  slowly  rising,  and  that  the  maximum  has  not  yet  been  reached. 
We  are  not  sure  whether  all  the  heat  produced  by  the  sun's  annual  shrinkage 
is  radiated  into  space,  or  whether  a  portion  is  retained  within  its  mass,  thus 
raising  its  temperature ;  or  whether,  again,  it  radiates  more  than  the  amount 
thus  generated,  so  that  its  temperature  is  slowly  diminishing. 

921.  That  the  sun  is  really  shrinking  is  admittedly  only  an  inference, 
for  the  shrinkage  must  be  far  too  slow  for  direct  observation.     Our  case  is 
like  that  of  a  man  who,  to  use  one  of  Professor  Newcomb's  illustrations, 
when  he  comes  into  a  room  and  finds  a  clock  in  motion,  concludes  that  the 
clock- weight  is  descending,  even  though  its  motion  is  too  slow  to  be  observed. 
Knowing  the  construction  of  the  clock  and  the  arrangement  of  its  gearing, 
and  the  number  of  teeth  in  each  of  its  different  wheels,  he  states  confidently 
just  how  many  thousandths  of  an  inch  the  weight  sinks  at  each  vibration  of 
the  pendulum ;  and  looking  into  the  clock-case  and  measuring  the  length  of 
the  space  in  which  the  weight  can  move,  and  noting  its  present  place,  he 
proceeds  to  state  how  long  ago  the  clock  was  wound  up,  and  how  long  it  has 
yet  to  run.     We  must  not  push  the  analogy  too  far,  but  it  is  in  some  such 
way  that  we  conclude  from  our  measurements  of  the  sun's  annual  output  of 
energy  in  the  form  of  heat,  how  fast  it  is  shrinking,  and  we  find  that  its 
diameter  must  diminish  not  far  from  200  feet  in  a  year ;  at  least,  the  loso  of 
potential  energy  corresponding  to  that  amount  of  shrinkage  would  account 
for  one  year's  running  of  the  solar  mechanism. 

922,  Age  of  the  Solar  System.  —  Looking  backward,  then,  in 
imagination  we  see  the  sun  growing  continually  larger  through  the 
reversed  course  of  time,  expanding  and  becoming  ever  less  and  less 
dense,  until  at  some  epoch  in  the  past  it  filled  all  the  space  now 
included  within  the  largest  orbit  of  the  solar  system. 

How  1  ng  ago  that  was  no  one  can  say  with  certainty.  If  we  could 
assume  that  the  amount  of  potential  energy  lost  by  contraction,  con- 
verted into  the  actual  energy  of  heat  and  radiated  into  space,  has 
been  the  same  each  year  through  all  the  intervening  ages,  and, 
moreover,  that  all  the  heat  radiated  has  come  from  this  source  only, 
without  subsidy  from  any  original  store  of  heat  contained  in  an 
original  "  fire  mist,"  or  from  energy  derived  from  outside  sources, 
then  it  is  not  difficult  to  conclude  that  the  sun's  past  history  must 
cover  some  15  000000  or  20  000000  years. 

But  the  assumption  that  the  loss  of  heat  has  been  even  nearly 
uniform  is  extremely  improbable,  considering  how  high  the  present 
temperature  of  the  sun  must  be  as  compared  with  that  of  the  original 


576  COSMOGONY. 

nebula,  and  how  the  ratio  of  surface  to  solid  content  has  increased 
with  the  lessening  diameter. 

Nor  is  it  unlikely  that  the  sun  may  nave  received  energy  from 
other  sources1  than  its  own  contraction.  Altogether  it  would  seem 
that  we  must  consider  the  15  000000  years  to  be  the  least  possible 
value  of  a  duration  which  may  hare  been  many  times  more  extended. 
If  the  nebular  hypothesis  and  the  theory  of  the  solar  contraction 
be  true,  the  sun  must  be  as  old  as  that, — how  much  older  no  one 
can  tell  with  certainty.  Lord  Kelvin,  however,  from  considerations 
based  mainly  on  the  observed  rise  of  temperature  downwards  from 
the  surface  of  the  earth,  and  the  heat-conducting  power  of  our  rocks 
is  disposed  to  set  a  maximum  limit  of  from  100  000000  to  200  000000 
years  for  the  possible  age  of  the  earth. 

It  is  precisely  here  that  the  nebular  hypothesis  encounters  its  most  serious 
difficulty.  It  would  seem  that  vastly  longer  periods  of  time  must  have  been 
required  for  the  formation  of  rings  and  nebulous  planets,  and  for  their  con- 
centration into  such  bodies  as  we  now  find  circulating  around  the  sun. 

923.  Future  Prospects.  —  Looking  forward  towards  the  future,  it 
is  easy  to  conclude  also  that  at  its  present  rate  of  radiation  and  con- 
traction the  sun  must,  within  5  000000  or  10  000000  years,  become 
so  dense  that  the  conditions  of  its  constitution  will  be  radically 
changed,  and  to  such  an  extent  that  life  on  the  earth,  as  we  now 
know  life,  would  probably  be  impossible.     If  nothing  intervenes  to 
reverse  the  course  of  things,  the  sun  must  at  last  solidify  and  become 
a  dark,  rigid  globe,  frozen  and  lifeless  among  its  lifeless  family  of 
planets.     At  least,  this  is  the  necessary  consequence  of  what  now 
seems  to  science  to  be  the  true  account  of  its  present  activity  and 
the  story  of  its  life. 

924.  Stars,  Star-Clusters,  and  the  Nebulae. — It  is  obvious  that  the 
same  nebular  hypothesis  applies  satisfactorily  to  the  explanation  of 
the  relation  of  these  different  classes  of  bodies  to  each  other.     In 
fact,  Herschel,  appealing  only  to  the  law  of  continuity,  had  concluded 
before  La  Place  formulated  his  theory,  that  nebulae  develop  some- 
times into  clusters,  sometimes  into  double  or  multiple  stars,  and 
sometimes  into  single  ones.    He  showed  the  existence  in  the  sky  of 
all  the  intermediate  forms  between  the  nebula  and  the  finished  star. 
For  a  time,  about  the  middle  of  our  century,  while  it  was  generally 

1  From  the  atomic  disintegration  of  radio-active  substances,  for  instance. 


THE    PRESENT    SYSTEM   NOT    ETERNAL.  577 

supposed  that  all  nebulae  were  nothing  but  star-clusters,  too  remote 
to  be  resolved  by  existing  telescopes,  his  views  fell  rather  into  abey- 
ance ;  but  when  the  spectroscope  demonstrated  the  substantial  differ- 
ences between  the  gaseous  nebulae  and  the  star-clusters,  they  regained 
acceptance  in  their  essential  features ;  with  perhaps  the  reservation, 
that  many  are  disposed  to  believe  that  the  rarest  even  of  nebulous 
matter,  instead  of  being  purely  gaseous,  is  full  of  solid  and  liquid 
particles  like  a  cloud  of  fog  or  smoke. 

925.  The  Present  System  not  Eternal. —  One  lesson  seems   to 
stand  out  clearly,  —  that  the  present  system  of  stars  and  worlds  is 
not  an  eternal  one.     We  have  before  us  irrefragable  evidence  of 
continuous,  uncompensated   progress,  inexorable  in  one  direction. 
The  hot  bodies  are  losing  their  heat,  and  distributing  it  to  the  cold 
ones,  so  that  there  is  a  steady,  unremitting  tendency  towards  a 
uniform  (and  therefore  useless)  temperature  throughout  the  universe  : 
for  heat  does  work,  and  is  available  as  energy  only  when  it  can  pass 
from  hotter  to  cooler  bodies,  so  that  this  warming  up  of  cooler  bodies 
at  the  expense  of  hotter  ones  always  involves  a  loss,  not  of  energy 
(for  that  is  indestructible),  but  of  available  energy.     To  use  the 
technical  language  now  usually  employed,  energy  is  unceasingly 
"dissipated"  by  the  processes  which  maintain  the  present  life  of 
the  universe ;  and  this  dissipation  of  energy  can  have  but  one  ulti- 
mate result,  —  that  of  absolute  stagnation  when  a  uniform  tempera- 
ture has  been  everywhere  attained.     If  we  carry  our  imagination 
backwards  we  reach  at  last  a  "  beginning  of  things,"  which  has  no 
intelligible  antecedent :  if  forwards,  an  end  of  things  in  stagnation. 
That  by  some  process  or  other  this  end  of  things  will  result  in 
"  new  heavens  and  a  new  earth  "  we  can  hardly  doubt,  but  science 
has  as  yet  no  word  of  explanation. 

926.  Sir  Norman  Lockyer's  Meteoritic  Hypothesis.  —  The  idea 
that  the  heavenly  bodies  in   their  present  state  may  have  been 
formed  by  the  aggregation  of  meteoric  matter,  rather  than  by  the 
condensation  of  a  gaseous  mass,  is  not  new,  and  not  original  with 
Mr.  Lockyer,  as    he   himself   points   out.      But  his  adoption  and 
advocacy  of  the  theory,  and  the  support  he  brings  to  it  from  spec- 
troscopic  experiments  on  the  light  emitted  by  fragments  of  meteoric 
stones  under  different  conditions,  has  given  it  such  currency  that 
his  name  will  always  be  justly  associated  with  it.    We  have  already 
referred  to  it  in  several  places  (Arts.  850  and  894  especially). 


578  LOCKYER'S  METEORITIC  HYPOTHESIS. 

He  believes  that  he  finds  in  the  spectra  of  meteorites,  under  vari- 
ous conditions,  an  explanation  of  the  spectra  of  comets,  nebulae, 
and  all  the  different  types  of  stars,  as  well  as  the  spectra  of  the 
Aurora  Borealis  and  the  Zodiacal  Light. 

Assuming  this,  he  considers  that  nebulae  are  meteoric  swarms  in 
the  initial  stages  of  condensation,  the  separate  individuals  being  still 
widely  separated,  and  collisions  comparatively  infrequent. 

As  aggregation  goes  on,  the  nebulae  become  stars,  which  run 
through  a  long  life-history,  the  temperature  first  increasing  slowly 
to  a  maximum,  and  then  falling  to  non-luminosity.  During  this  life- 
history  the  stars  pass  through  successive  stages,  each  stage  char- 
acterized by  its  own  typical  spectrum.  Lockyer  has  also  proposed 
an  elaborate  classification  of  stellar  spectra  arranged  according  to 
these  hypothetical  stages ;  but  it  has  not  yet  secured  general 
acceptance,  probably  because  its  theoretical  basis  appears  to  be 
insufficiently  established. 

The  hypothesis  receives  a  certain  support  from  a  most  interesting 
mathematical  investigation  of  Prof.  George  Darwin,  who  shows 
that,  if  we  assume  a  meteoric  swarm  comparable  in  dimensions  with 
our  solar  system,  composed  of  individual  masses  such  as  fall  on  the 
earth,  and  endowed  with  such  velocities  as  meteors  are  known  to 
have,  such  a  swarm,  seen  from  the  distance  of  the  stars,  would 
behave  like  a  mass  composed  of  a  continuous  gas.  This  is  not  strange, 
since,  according  to  the  kinetic  theory  of  gases,  a  gas  is  simply  a 
swarm  of  molecules,  behaving  in  just  the  way  the  meteorites  are 
supposed  to  act. 

926*.  The  Planetesimal  Hypothesis. —  This  is  a  new  form  of  the 
meteoric  theory,  recently  proposed  and  developed  by  Chamberlin 
and  Moulton.  It  assumes  as  the  origin  of  the  solar  system,  a 
spiral  nebula,  composed  largely  of  little  masses  (planetesimals) 
moving  around  the  centre,  generally  in  the  same  direction,  but  in 
orbits  that  vary  in  inclination,  eccentricity,  and  period,  and  are 
subject  to  continual  perturbation.  There  results  a  very  slow  accre- 
tion of  the  planetesimals  into  planets,  with  very  little  development 
of  heat,  since  the  relative  velocities  of  the  colliding  bodies  are 
very  small. 

A  great  advantage  of  this  theory  is  that  it  allows  time  enough  to 
satisfy  the  most  exorbitant  demands  of  geology  and  biology. 


EXERCISES.  579 

EXERCISES  ON  CHAPTER  XXII. 

1.  Find  the  mass  of  the  system  of  Alpha  Centauri  from  the  data  given 
in  Tables  IV.  and  V.  —  namely,  parallax  (p)  =  0".75,  semi-major  axis  of 
orbit  (a")  =  17".70,  and  period  (t)  =  81.1  years. 

Ans.    Mass  of  system  =  2.00  X  mass  of  the  sun. 

2.  Find  the  mass  of  the  system  of  Sirius  from  the  tabular  data. 

Ans.   2.78  X  mass  of  the  sun. 

3.  Find  the  mass  of  the  system  of  Eta  Cassiopeise  from  the  tabular  data. 

Ans.    0.34  X  mass  of  the  sun. 

4.  Find  the  mass  of  the  system  of  70  Ophiuchi  from  the  tabular  data. 

Ans.   0.77  X  mass  of  the  sun. 

5.  Find  the  radius  of  the  apparent  orbit  of  the  spectroscopic  binary  3105 
Lacaille,  the  relative  velocity  of  the  components  being  385  miles  a  second, 
and  the  period  3  days,  2  hours,  and  46  minutes,  as  indicated  by  the  doubling 
of  the  lines  in  the  spectrum.     Assume  that  the  orbit  is  circular,  that  its 
plane  is  directed  towards  the  sun,  and  that  the  two  components  are  equal. 

Ans.   Radius  of  orbit  =16  493000  miles. 

6.  Compute  the  mass  of  the  system  on  the  same  assumptions  as  above, 
remembering  that  the  radius  of  this  apparent  orbit  is  also  the  radius  of  the 
relative  orbit  which  each  component  describes  around  the  other  regarded  as 

atrestf  Ans.   76.75  x  mass  of  the  sun. 

7.  Carry  out  similar  computations  for  the  systems  of  Zeta  Ursae  Majoris, 
Beta  Aurigse,  and  Mu  Scorpii,  using  the  data  of  Art.  879. 

8.  Determine  the  radius  of  the  orbit  described  by  Spica  Virginis,  as 
shown  by  the  shift  of  the  lines  in  its  spectrum.     Velocity  =56.6  miles  a 
second  ;  period  =  4  days  and  19  minutes.     Orbits  assumed  circular  and  in 
plane  of  the  sun.  ^    Radiug  = 


9.  From  this  determine  the  mass  of  the  system,  assuming  that  the  mass 
of  the  bright  star  is  infinitesimal  as  compared  with  that  of  the  dark  star  ; 
i.e.,  that  it  is  a  small  planet  revolving  around  a  dark  central  sun.     (A  very 
improbable  hypothesis  of  course.) 

Ans.   0.315  X  mass  of  the  sun. 

10.  What  is  the  mass  of  the  system  if  the  dark  star  is  equal  to  the  bright 
one?     (In  this  case  the  radius  of  the  relative  orbit  is  the  diameter  of  the 
apparent  orbit  of  Spica,  or  double  its  value  in  the  last  example.) 

Ans.    8  X  0.315,  or  2.520  X  mass  of  the  sun. 


580  EXERCISES. 

11.  What  is  the  mass  if  the  dark  star  has  a  mass  only  one-fourth  that  of 
the  bright  one?  (In  this  case  the  orbit  of  the  dark  star  has  a  radius  four 
times  as  great  as  that  of  Spica,  and  the  radius  of  the  relative  orbit  is  five 
times  as  great  as  that  of  the  apparent  orbit  of  Spica.) 

Ans.  125  X  0.315,  or  39.37  X  mass  of  the  sun  ;  the  mass  of  the 
bright  star  being  31.50,  arid  that  of  the  dark  star  being  one-fourth 
as  great,  or  7.87. 

NOTE.  —  The  assumption  that  the  bright  star  is  a  mere  planet,  revolving  around  a  dark 
central  body  vastly  more  massive  than  itself,  gives  us  a  minor  limit  to  the  possible  mass  of 
the  system,  but  the  major  limit  cannot  be  fixed  without  knowledge  as  to  the  relative  mass 
of  the  dark  body. 

If  the  dark  body  is  larger  than  the  bright  one,  the  mass  of  the  system  cannot  exceed  eight 
times  that  minor  limit. 

The  general  formula  is  easily  obtained  :  let  n  be  the  ratio  between  the  masses  of  the  bright 
and  dark  stars,  so  that  if  r  is  the  radius  of  the  circle  described  by  the  bright  star  around  the 
common  centre,  the  radius  of  the  circle  described  by  the  other  will  be  nr,  and  the  radius  of  the 
relative  orbit  will  be  (n  +  1)  r.  Also  let  /u.  be  the  united  mass  of  the  two  stars.  Then,  express- 
ing the  period,  t,  in  years,  r  in  astronomical  units,  and  ft.  in  terms  of  the  sun's  mass,  we  have 


The  factor  (n  +  I)3  becomes  unity  when  n=  o,  i.e.,  when  the  bright  star  is  a  particle  ;  and 
infinity  when  n  becomes  infinite,  i.e.,  when  the  dark  star  is  a  particle  revolving  at  the  infinite 
distance,  r(n  +  1).  It  becomes  8  when  n=l,  the  two  stars  being  equal. 

It  may  be  added  that  the  assumption  that  the  orbit  is  circular,  and  that  its  plane  passes 
through  the  solar  system,  is  entirely  gratuitous  and  not  likely  to  be  correct.  But  the 
general  character  of  the  results  would  not  be  seriously  changed  unless  the  inclination  and 
eccentricity  of  the  orbit  were  great. 

NOTE  TO  ART.  879. 

The  number  of  spectroscopic  binaries  recently  detected  by  various  observers 
in  this  country  and  Europe  is  so  large  that  it  is  useless  to  try  to  keep  up  the 
enumeration  in  this  text-book.  The  list  at  present  (January,  1906)  includes  about 
200,  and  is  growing  continually.  Professor  Campbell  of  the  Lick  Observatory 
who  has  been  specially  successful  in  this  line  of  work,  estimates  that  about  one 
star  in  every  twelve  examined  turns  out  a  "spectroscopic  binary."  Perhaps 
the  two  most  notable  additions  to  the  catalogue  are  Polaris  and  Capella,  —  the 
former  with  a  period  a  little  less  than  four  days,  and  a  range  of  radial  velocity 
of  only  about  four  miles  a  second  ;  the  other  with  a  period  of  105  days  and  a 
velocity  range  of  about  36  miles.  The  actual  orbital  velocity  cannot,  of  course, 
be  determined  until  we  know  the  inclination  of  the  orbit  to  the  line  of  sight. 


ADDENDUM   A.  580  a 


ADDENDUM   A. 

THE  SPECTROHELIOGRAPH   AND   ITS    APPLICATION   TO   SOLAR 
RESEARCH.       THE   SIDEROSTAT   AND   COSLOSTAT. 

THE  spectroheliograpli  received  a  passing  mention  in  Art.  326*, 
but  it  has  recently  become  so  important  that  something  more  is  now 
required.  The  principle  of  the  instrument,  first  suggested  by  Jans- 
sen  in  1870,  was  first  successfully,  and  independently,  applied  by 
Hale  in  Chicago,  and  Deslandres  in  Paris,  about  1890.  Its  essential 
feature  is  the  introduction  in  a  photographic  spectroscope  of  a  second 
slit,  parallel  to  the  collimator  slit,  but  at  the  other  end- of  the  instru- 
ment, close  in  front  of  the  sensitive  plate,  thus  isolating  a  narrow 
line  of  homogeneous  light  in  the  spectrum.  If  now  the  telescope 
which  carries  the  spectroscope  is  pointed  at  the  sun  and  its  image 
made  to  pass  over  the  collimator  slit,  while  at  the  same  time  the 
sensitive  plate  itself  is  moved  in  precisely  corresponding  manner, 
the  result  will  be  a  photograph  of  whatever  passed  over  the  slit, 
produced  solely  by  the  light  of  that  single  wave-length  which  was  iso- 
lated by  the  second  slit. 

If,  for  instance,  the  spectroscope  were  so  adjusted  as  to  bring 
upon  the  second  slit  the  K  line  of  calcium,  then  the  image  photo- 
graphed would  be  due  to  calcium  vapor  only,  and  that  vapor  in  such 
physical  condition  as  to  emit  this  " K  light,"  if  the  expression  may 
be  permitted.  If  the  adjustment  were  such  as  to  bring  a  line  of 
hydrogen,  magnesium,  or  iron  into  the  slit,  we  should  get  an  image 
due  solely  to  the  corresponding  element.  In  order  that  an  image  of 
the  entire  sun  be  obtained  it  is  of  course  necessary  that  the  two  slits 
should  be  longer  than  the  diameter  of  the  sun's  image  upon  the  slit- 
plate. 

Other  arrangements  of  the  instrument  are  possible.  In  that  first 
used  by  Hale  the  telescope  was  kept  directed  at  the  centre  of  the 
sun  and  the  sensitive  plate  was  fixed ;  but  the  two  slits  were  mov- 
able and  so  connected  by  a  system  of  levers  that  while  the  first  was 
made  to  traverse  the  sun's  image,  the  second  was  made  to  move 
correspondingly  before  the  plate.  But  this  arrangement  is  much 
more  difficult  to  construct  and  adjust,  and  is  in  other  ways  inferior. 


580  b  ADDENDUM   A. 

Still,  at  Kenwood,  between  1892  and  1896,  a  number  of  beautiful 
photographs  of  prominences  was  thus  obtained,  and  many  interest- 
ing plates  showing  the  calcium  floccules  over  the  entire  disc. 

The  Rumford  spectroheliograph,  now  used  on  the  great  Yerkes 
telescope,  has  slits  6  inches  long,  nearly  but  not  quite  sufficient  to 
take  in  the  whole  diameter  of  the  sun's  image  formed  by  the  40-inch 
telescope.  The  slit  is  placed  east  and  west,  and  the  telescope  is 
moved  in  declination  by  a  slow-motion  screw  driven  by  an  electric 
motor,  this  motion  being  also  communicated  directly  to  the  photo- 
graphic plate. 

It  must  be  remembered  that  the  dark  lines  of  the  solar  spectrum 
are  dark  only  relative  to  the  background;  they  are  really  bright,  and 
somewhat  brighter  than  the  same  lines  in  the  "  flash  spectrum,"  for 
their  actual  brilliance  is  not  reduced  by  the  light  which  reaches  the 
reversing  layer  from  the  photosphere  beneath,  but  somewhat  in- 
creased (Art.  314,  note).  When,  therefore,  the  second  slit  is  set  on  a 
"dark  line"  of  the  solar  spectrum,  the  background  of  continuous 
spectrum  being  excluded,  an  observer  looking  through  an  eye-piece 
would  see  the  line  bright  and  capable  of  impressing  itself  upon  a 
sensitive  film.  The  instrument  therefore  makes  it  possible  to  study 
separately  and  in  detail  what  may  be  called  the  calcium  surface, 
the  hydrogen  surface,  the  magnesium  surface,  etc.,  of  the  sun.  And 
in  the  differences  and  correspondences  of  these  surfaces  there  is  a 
mine  of  important  information. 

In  the  new  solar  observatory  which  Professor  Hale,  with  the 
subvention  of  the  Carnegie  Institution,  has  recently  erected  upon 
Wilson's  Peak  in  Southern  California,  a  fixed  horizontal  telescope 
fed  by  a  "  coelostat "  is  now  in  use.  A  large  image  of  the  sun  is 
obtained,  which  may  be  photographed  directly  or  may  be  used  for 
the  corresponding  spectroheliograph  firmly  mounted  on  piers.  Sev- 
eral other  spectroheliographs  of  smaller  size  and  different  construc- 
tions are  being  provided  for  stations  in  Europe  and  India,  which 
will  cooperate  in  solar  studies,  and  important  results  are  already 
following  from  the  data  obtained. 

In  various  lines  of  astrophysical  work  there  are  great  advantages 
in  having  the  telescope  fixed  in  a  convenient  position  with  the 
object  under  observation  reflected  into  it  steadily  by  a  mirror  suit- 
ably moved  by  clockwork. 

The  Siderostat  of  Eoucault  (1865)  consists  of  a  plane  mirror  car- 
ried by  a  polar  axis  which  revolves  once  in  24  hours.  The  mirror 
can  thus  be  so  set  as  to  throw  the  reflected  light  in  any  chosen 


ADDENDUM   A. 


580  0 


FIG.  247.  —  Oct.  9,  3  h.  30  m.    Calcium  Flocculi.    Slit  set  on  H. 


FIG.  248.  —  Oct.  9,  1  h.  04  m.     Hydrogen  Flocculi.     Slit  set  on 


THE  GREAT  SUN-SPOT  OF  OCTOBER,  1903. 

Scale  :  Sun's  Diameter  =  9jf  inches. 


580  d  ADDENDUM   A. 

direction  (usually  horizontally  north  or  south),  and  means  are  pro- 
vided by  which  the  observer  can  adjust  it  from  his  distant  position. 
This  is  perfectly  satisfactory  for  all  work  which  requires  merely  a 
fixed  and  convenient  direction  for  the  central  line  of  the  reflected 
beam  —  as  for  the  study  of  star-spectra  or  the  general  spectrum  of 
the  sun.  If,  however,  the  instrument  is  used  to  form  an  image 
of  the  sun  or  of  a  group  of  stars,  it  has  the  serious  disadvantage 
that  the  image  revolves  around  its  central  point  and  with  a  speed 
that  continually  varies,  thus  rendering  the  arrangement  unsuitable 
for  most  photographic  work. 

The  Coelostat  differs  in  that  the  polar  axis  which  carries  its  mir- 
ror turns  once  in  48  hours  instead  of  24.  The  reflected  image  in 
this  case  does  not  revolve,  but  the  possible  directions  in  which  it  can 
be  thrown  are  not  arbitrary,  being  determined  by  the  declination  of 
the  object.  The  difficulty  can  be  overcome  by  using  a  second  mirror 
to.  send  the  reflected  beam  when  it  is  wanted,  but  of  course  this  is 
expensive  and  involves  loss  of  light.  This  arrangement,  however, 
is  proposed  for  Mt.  Wilson,  since  a  non-rotating  image  is  essential, 
as  is  also  the  possibility  of  mounting  the  spectroheliograph  and  its 
accessories  upon  solid  piers. 

As  an  interesting  specimen  of  the  photographs  obtained  by  the 
spectroheliograph  we  give  Figs.  247  and  248,  reduced  by  one-thrd, 
from  Plate  XI  of  Professor  Hale's  paper  describing  the  instrument 
and  its  work.  The  upper  figure  of  the  two  is  a  "  calcium "  photo- 
graph of  a  great  sun  spot ;  the  lower,  a  "  hydrogen  "  photograph  of 
the  same,  both  taken  the  same  afternoon.  The  differences  are 
instructive,  especially  the  peculiarly  "  wispy  "  character  of  the  dark 
markings  of  the  hydrogen  plate  as  compared  with  the  "  lumpiness  " 
of  the  calcium  floccules. 


ADDENDUM  B.  5800 


ADDENDUM  B. 

NOVA  PERSEI  AND  NOVA  GEMINORUM. 

A  REMARKABLE  temporary  star,  the  most  brilliant  since  Kepler's 
star  of  1604,  suddenly  appeared  in  1901,  probably  on  February  20, 
though  first  seen  (by  Dr.  Anderson  in  Edinburgh)  on  the  21st,  when  it 
was  about  as  bright  as  Polaris.  Photographs  covering  the  region  of 
the  star  made  at  Cambridge,  United  States,  on  several  dates  preced- 
ing and  including  the  19th,  show  that  on  the  19th  the  star  was  not 
yet  as  bright  as  the  12th  magnitude.  Within  three  days  it  increased 
its  brightness  fully  25,000  fold,  and  on  the  22d  was  for  several  hours 
the  brightest  star  in  the  heavens,  Sirius  alone  excepted,  and  more 
than  a  match  for  Capella  and  Vega.  It  was  then  distinctly  ruddy, 
very  like  Arcturus.  It  faded  at  first  rapidly  but  fitfully,  and  by  the 
end  of  March  was  barely  visible  to  the  naked  eye;  after  that  its 
decrease  was  more  gradual,  and  it  is  still  (1908)  visible  in  large  tele- 
scopes as  a  little  star  of  12th  or  13th  magnitude. 

The  spectrum,  as  photographed  at  Cambridge  on  the  22d,  was  not 
that  of  the  usual  "Nova"  type,  but  much  resembled  that  of  a  Orionis 
(Rigel),  being  mainly  continuous,  though  crossed  by  about  30  not  very 
conspicuous  dark  lines.  Clouds  prevented  photographs  on  the  23d, 
but  on  the  24th  it  was  clear,  and  in  the  meantime  a  complete 
change  had  taken  place.  The  spectrum  was  now  essentially  like 
that  of  Nova  Aurigse,  with  the  same  broad  bright  bands  of  hydrogen 
and  their  dark  and  more  refrangible  companions.  It  may  be  noted 
here  that  the  recent  (1903)  investigations  by  Ebert  of  Munich  go  far 
towards  proving  that  for  the  explanation  of  these  doubled  lines  we 
need  not  resort  to  the  hypothesis  of  conflicting  masses  of  hydrogen, 
cool  and  hot,  moving  towards  and  from  us  with  a  speed  of  several 
hundred  miles  a  second,  nor  even  to  explosive  pressures.  It  seems 
probable  that  the  phenomena  may  be  due  merely  to  "anomalous 
refraction"  in  portions  of  the  star's  gaseous  envelope  under  power- 
ful compression  and  intense  luminous  excitement. 

Since  then  the  spectrum  has  followed  the  usual  course,  having 
become  nebular  before  the  end  of  the  year,  though  with  some  non- 
nebular  peculiarities  in  the  extreme  width  of  its  bright  hydrogen 


580/  ADDENDUM    B. 

lines  and  in  the  presence  of  some  conspicuous  lines  not  yet  found 
in  nebulae.  It  does  not  yet  appear  whether  its  spectrum  will  finally 
revert  to  the  purely  continuous,  stellar  type  like  that  of  Nova  Aurigae. 

During  the  star's  decline  its  brightness  oscillated  as  much  as  a 
whole  magnitude,  the  irregular  interval  between  maxima  ranging 
from  about  two  days  in  February  to  six  or  eight  in  the  autumn. 

In  September  it  became  possible  to  photograph  the  invisible 
nebulosity  around  it  with  the  reflectors  (not  the  great  refractors)  of 
the  Lick  and  Yerkes  observatories.  It  was  found  to  be  very  exten- 
sive, roughly  circular,  with  an  apparent  diameter  about  half  that  of 
the  moon;  and  since  the  most  careful  observations  have  been  unable 
to  detect  any  parallax  or  proper  motion  of  the  star,  it  is  clear  that 
its  distance  exceeds  that  of  any  of  the  nearer  stars,  —  very  likely  it 
is  as  much  as  100  light  years,  and  not  improbably  greater  yet.  If 
so,  the  diameter  of  the  nebula  must  have  been  at  least  1400  times 
that  of  the  earth's  orbit,  and  this  is  probably  an  underestimate. 

There  were  in  it  several  pretty  well-defined  knots  and  streaks  of 
condensation,  and  the  photographs  soon  brought  to  light  an  almost 
astounding  phenomenon.  These  knots  were  found  to  be  all  moving 
swiftly  away  from  the  star  at  various  rates  averaging  about  10"  a 
week,  —  a  motion  apparently  not  very  rapid  as  seen  from  the  earth. 
But  if  the  Nova  were  as  near  as  our  next  neighbor,  a  Centauri,  this 
would  mean  more  than  two  thousand  miles  a  second;  not  improbably 
the  distance  is  a  hundred  times  as  great,  and  if  so,  the  speed  becomes 
greater  than  that  of  light  itself.  Indeed,  the  most  plausible  explana- 
tion of  the  phenomenon  yet  offered  is  that  suggested  by  Kapteyn,  — 
that  the  motion  is  only  apparent,  not  an  actual  rush  of  masses 
of  matter,  but  simply  the  progressive  illumination  of  spiral  streams 
of  nebulosity,  advancing  along  them  with  the  speed  of  light,  an 
illumination  originating  when  the  star  first  flashed  out.  If  this 
explanation  is  correct,  the  distance  of  the  star  must  be  about  300 
light  years,  and  the  actual  outburst  occurred  about  the  time  when 
Columbus  was  discovering  America. 

Another  small  "Nova"  was  discovered  in  Gemini  in  January,  1903, 
by  Professor  Turner  of  Oxford  in  examining  star-chart  photographs 
there  made.  It  was  of  only  the  8th  magnitude  and  presented 
nothing  of  special  interest. 

It  is  a  notable  and  perhaps  significant  fact  that  without  exception 
all  the  temporary  stars  thus  far  observed  have  been  in  or  near  the 
Milky  Way.  It  will  be  remembered  that  the  same  is  true  of  all  the 
Wolf-Eayet  stars. 


ADDENDUM   C.  580  # 


ADDENDUM   C. 

SUPPLEMENTARY  TO   ARTICLES   847,    848,   AND   852. 

(To  Art.  847.)  The  short-period  variables  of  the  y  Aquilse  and 
/8  Lyrse  type  are  mostly  "  punctual  variables,"  to  use  Miss  Clerke's 
expression ;  i.e.,  like  the  Algol  variables,  their  periods  are  uniform 
without  any  such  irregularities  as  are  usual  with  the  stars  of  the 
o  Ceti  type.  It  is  natural  to  ascribe  such  "  punctuality "  to  an 
orbital  revolution,  and  this  is  justified  in  many  cases  by  the  fact 
that  the  star  is  proved  to  be  a  spectroscopic  binary  by  the  periodical 
shifting  or  doubling  of  its  spectrum-lines.  But  the  changes  of 
brightness  cannot,  as  with  the  Algol  stars,  be  explained  "  geometri- 
cally," by  eclipses;  while  the  periods  are  regular,  the  times  of  maxima 
and  minima  do  not  accord  with  such  an  explanation,  and  besides,  the 
changes  are  continuous,  and  not  confined  to  any  short  portion  of 
the  period.  It  seems  necessary  to  suppose,  therefore,  that  the  orbi- 
tal revolution  is  accompanied  by  some  mutual  action  of  the  two  (or 
more)  revolving  stars  upon  each  other  (possibly  tidal,  possibly  some 
kind  of  physical  influence)  and  that  this  action  causes  fluctuations 
in  the  radiating  power  of  their  surfaces. 

But  precise  explanation  is  still  difficult  and  uncertain,  and  very 
possibly  no  one  explanation  will  apply  to  all  the  stars  now  grouped 
together  in  this  class. 

The  great  majority  of  these  variables,  like  the  temporary  stars 
and  the  Wolf-Rayet  stars,  are  near,  or  in,  the  Milky  Way.  This  is 
not  true  of  the  stars  of  the  o  Ceti  or  Algol  types,  which  are  found 
indiscriminately  in  all  parts  of  the  heavens. 

(To  848.)  An  interesting  fact  respecting  the  "  eclipse  stars  "  of 
the  Algol  type  was  first  pointed  out  in  1899,  simultaneously  but 
independently,  by  Russell  at  Princeton  and  by  Roberts  of  Lovedale, 
South  Africa;  namely,  that  an  upper  limit  to  the  density  of  the 
eclipsing  stars  can  be  determined  from  the  ratio  of  the  whole  period 
of  the  variable  to  the  time  between  the  beginning  and  the  end  of 
the  obscuration.  In  the  cases  of  the  seven  or  eight  Algol  stars,  for 
which  we  now  have  sufficiently  accurate  data,  it  was  found  that  this 
upper  limit  is  far  below  the  density  of  the  sun,  and  of  course  the 


580  h  ADDENDUM   C. 

actual  density  lower  yet :  some  of  them  must  be  hardly  more  sub- 
stantial than  clouds. 

(To  852.)  Since  1900  the  catalogue  of  known  variables  has 
increased  enormously.  The  sporadic  variables  (excluding  the  hun- 
dreds which  have  been  found  in  star-clusters,  in  the  Magellanic 
clouds,  and  in  a  region  of  some  square-degrees  including  the  Nebula 
of  Orion)  must  now  number  nearly  1000,  if  not  more  ;  and  the  list  is 
continually  and  rapidly  growing. 

The  discoveries  are  now  made  mostly  by  photography,  partly  from 
negatives  made  specially  for  the  purpose  of  detecting  variables,  and 
partly  in  the  examination  of  the  photographs  of  the  star-charting 
campaign.  Numerous  amateur  observers  devote  themselves  to  work 
in  this  line,  and  at  several  observatories  the  accurate  study  of  the 
light-curves  of  variables  with  large  telescopes  and  elaborate  pho- 
tometric apparatus  forms  a  considerable  part  of  the  work  of  the 
institution. 

Professor  Pickering  in  his  observatory  report,  dated  Sept.  30, 1905,  stated 
that  since  the  Harvard  photographic  work  began  in  1886,  2750  variables 
have  been  discovered,  —  2197  at  Cambridge,  and  about  555  elsewhere. 
Mrs.  Fleming  has  discovered  8  "  novae  "  and  197  variables,  mainly  by  bright 
hydrogen  lines  in  their  spectra ;  Professor  Bailey  has  detected  509  in  globu- 
lar star-clusters  ;  and  Miss  Leavitt  1442,  mostly  in  and  near  the  Magellanic 
clouds.  And  since  that  date  the  list  has  been  considerably  lengthened. 

Of  course  nearly  all  of  these  new  variables  are  extremely  faint,  observable 
only  by  great  telescopes  or  by  photography;  and  for  the  great  majority 
nothing  is  yet  known  as  to  the  period  and  type  of  variation. 

The  total  number  of  variables  which  can  be  reached  by  our  present  instru- 
ments must  be  hundreds  of  thousands,  and  not  improbably  millions.  Among 
the  6000  naked-eye  stars  about  70  variables  are  already  known,  and  there  is 
no  reason  to  suppose  that  the  proportion  is  different  for  the  telescopic  stars. 


ADDENDUM   D  580 


ADDENDUM    D. 

ON  ASTEROIDS. 

THE  present  rate  of  asteroid  discovery  makes  it  impossible  to 
keep  up  with,  it  in  a  text-book.  In  1906  more  than  a  hundred  were 
found  (the  majority  at  Heidelberg),  and  1907  even  exceeded  that 
record.  Most  of  them  are  so  faint  as  to  be  observable  only  by 
photography,  although  one  of  magnitude  9.2  was  discovered  in  1908. 

Rev.  J.  S.  Metcalf  of  Taunton,  Massachusetts,  is  using  a  very 
effective  modification  of  the  Heidelberg  method.  While  the  tele- 
scope follows  the  stars  by  its  driving-clock,  the  photographic  plate 
receives  a  slight  sliding  motion  equal  to  that  of  an  average  asteroid 
in  the  region  under  observation.  The  image  of  a  planet,  if  one 
happens  to  be  present,  remains  therefore  stationary  on  the  plate,  or 
nearly  so,  during  the  entire  exposure,  and  is  many  times  more  in- 
tense than  if  it  had  been  allowed  to  trail.  The  stars  do  the  trailing, 
and  are  readily, recognized  as  such. 

When  first  announced  each  object  is  designated  provisionally  by 
an  alphabetical  arrangement  which  began  with  AA,  in  1893;  thus 
Eros  was  for  a  time  known  as  DQ.  The  combination  of  ZZ  was 
reached  in  1907,  but  the  same  system  is  being  continued,  always, 
however,  prefixing  the  year,  as  "  1908  C J."  When  sufficient  obser- 
vations have  been  made  to  determine  the  planet's  orbit  and  its 
non-identity  with  any  previously  known,  the  Director  of  the  Rechen- 
institut  at  Berlin  assigns  a  permanent  "  number,"  and  the  discoverer, 
if  he  chooses,  gives  it  a  name. 

Among  the  recently  discovered  planets,  TG  (588),  1906  VY,  and 
1907  XM  (all  discovered  at  Heidelberg)  are  of  special  interest.  They 
form  a  class  by  themselves,  their  orbits  not  differing  greatly  from 
that  of  Jupiter  in  size  and  period.  They  have  already  received  the 
names  of  Achilles,  Patroclus,  and  Hector.  The  problem  of  their 
motion  is  peculiarly  interesting,  for  they  appear  to  present  approxi- 
mately the  special  case  long  ago  pointed  out  by  Lagrange,  of  a  planet 
keeping  permanently  equidistant  from  the  sun  and  Jupiter. 

The  Asteroid  Ocllo  (475),  discovered  in  1901,  has  an  eccentricity 
of  0.38,  even  greater  than  that  of  JEthra.  Planet  1906  WD  has  the 
enormous  inclination  of  48°. 


APPENDIX. 


SUPPLEMENTARY  ARTICLES. 

1000.  Reduction  of  Sidereal  Time  to  Mean  Solar  Time  and 
Vice  Versa.  (Supplementary  to  Art.  112.)  —  Since  the  tropical  year 
(Art.  216)  contains  365.2421  mean  solar  days,  and  exactly  one 
more  sidereal  day,  it  follows  that  the  number  of  sidereal  seconds  in 
any  time  interval  is  equal  to  the  number  of  mean  solar  seconds 


multiplied  by  r,  *.&  by  1.00273791.     Hence,  if  /and  /'  are 

ODO.^4^1 

respectively  the  number  of  mean  solar  and  sidereal  seconds  in  any 
time  interval,  we  have  /'  =  /+  0.00273791  /.  (The  logarithm 
of  the  decimal  coefficient  is  [7.4374191]).  Also, 

=  r  x  0.9972696  =  I'  —  0.00273043  /'. 

The  log.  of  the  coefficient  is  [7.4362311]. 

The  "  American  Ephemeris  "  gives  at  the  end  of  the  book  two 
tables  containing  the  values  of  the  second  terms  of  the  two 
formulae  for  every  value  of  /  and  /'  up  to  24  hours,  and  the 
reduction  of  any  sidereal  interval  to  solar,  or  the  reverse,  is  accom- 
plished by  simply  adding  or  subtracting  the  tabular  correction. 

To  reduce  a  given  moment  of  sidereal  time  to  solar,  or  the 
reverse,  we  must  first  have  the  local  sidereal  time  of  the  preceding 
mean  solar  noon.  This  "  sidereal  time  of  mean  noon  "  is  given  in 
the  Almanac  for  every  day,  and  for  the  meridians  of  Washington 
and  Greenwich.  It  must  be  corrected  for  the  longitude,  A,  of  the 

observer  by  applying  the  correction  2358.91  X  7^1  >  which  may  be 

taken  directly  from  the  table  for  reducing  sidereal  intervals  to  solar, 
using  A  as  the  argument  ;  the  correction  is  to  be  added  for  West 
longitude,  subtracted  for  East. 

To  convert  sidereal  time  into  solar,  we  subtract  from  the  given 
time  the  sidereal  time  at  preceding  noon  (duly  corrected  for  longi- 


^3 

- 
582  APPENDIX. 

tilde).  The  difference  is  the  interval  since  noon,  expressed  in 
sidereal  units,  and  is  at  once  reduced  to  solar  time  units  by  sub- 
tracting the  correction  taken  from  the  proper  almanac  table 
(Table  II).  If  the  given  sidereal  time  is  numerically  smaller 
than  the  time  at  preceding  mean  noon,  24  hours  must  be  added. 

If  the  given  time  is  solar,  we  add  it  to  the  sidereal  time  at 
preceding  mean  noon,  and  also  add  the  correction  from  the  other 
almanac  table  (Table  III),  which  gives  the  reduction  of  solar  time 
intervals  to  sidereal. 

Example:  Reduce  to  local  mean  solar  time  the  sidereal  time 
18h33m278.71  at  Princeton,  N.  J.  (A.  =  9ra328.6  east  of  Washington), 
on  Aug.  27,  1896. 

h.     m.       s. 

Sidereal  time  of  mean  noon  at  Washington  10  25  43.03 

Reduction  for  longitude  (Aim.  p.  528)  —1.57 

Princeton  sidereal  time  of  noon  10  25  41.46 

Sidereal  time  given  18  33  27.71 

Sidereal  interval  since  noon  8  07  46.25 
Reduction  to  mean  time  interval  (p.  529)  (Table  II)         —  1  19.91 

Corresponding  mean  time  (local)  8  06  26.34 

If  "  standard  time "  is  wanted  (Art.  122)  we  must  further  correct 
the  local  mean  solar  time  by  subtracting  lm  22s. 50,  the  difference 
between  five  hours  and  the  longitude  of  Princeton.  Conversely,  to 
reduce  this  mean  time,  8h06m268.34,  to  sidereal  we  proceed  as 
follows  : 

h.     m.        s. 

Sidereal  time  of  Princeton  noon  10  25  41.46 

Given  mean  time  8  06  26.34 

Reduction  to  sidereal  (p.  532)  (Table  III)    +  1  19.91 
Sidereal  time  18  33  27.71 

A  rough  reduction,  usually  correct  within  five  or  ten  minutes,  is  easily  made 
without  any  computation  or  tables  by  assuming  that  the  sidereal  time  at  noon 
increases  uniformly  two  hours  each  calendar  month,  reckoning  from  March 
21st;  passing  the  even  hours  on  the  21st  of  each  month  and  the  odd  hours  on 
the  6th.  For  days  between,  allow  4m  each,  and  108  additional  for  each  hour 
elapsed  since  noon.  Thus,  taking  the  example  above,  we  have  for  the  Princeton 
sidereal  time  of  noon  on  Aug.  27th  10h  24m. 

Princeton  sidereal  time  of  noon 

Given  sidereal  time 

Sidereal  interval 

Correction 

Corresponding  mean  time 
The  result,  however,  is  seldom  quite  so  nearly  correct,  as  the  method  takes  no 
account  of  the  equation  of  time,  or  the  inequality  of  the  months. 


APPENDIX. 


583 


FIG.  240. 


1001.  Azimuthal  Motion  of  a  Star  at  the  Horizon.  (Supple- 
mentary to  Art.  141.)  —  In  Fig.  240  SS1  is  part  of  the  diurnal  circle 
of  a  star  that  is  rising,  the  arc  SS1  being  described  in  a  unit  of  time, 
—  say,  one  minute.  Then  ab,  the  corresponding  arc  of  the  equator, 
is  15',  and  SS',  expressed  in  minutes  of  a  great  circle,  is  15'  cos  8,  8 
being  the  star's  declination  aS. 

In  passing  over  SS1  its 
azimuth  changes  by  the  arc 
SM,  which  (from  the  relations 
in  the  small,  sensibly  plane 
triangle  SS'M)  equals 
SS'  X  cos  S'SM 

=  15'  cos  8  X  cos  S'SM.  (1) 

Now  S'SM=ZSP  in  the 
ZPS  spherical  triangle,  and 
in  this,  by  the  fundamental 
equation  of  spherical  trigonometry,  we  have 

cos  PZ=  cos  ZS  X  cos  PS—  sin  ZS  X  sin  PS  X  cos  ZSP. 
Substituting  the  astronomical  values,  this  gives 

sin  <j>  =  cos  £  sin  8  —  sin  £  cos  8  X  cos  ZSP. 

Whence          cos  ZSP  (or  SS'M)  =  Bin* -cog  {sin  8. 

sm  £  cos  8 

When  the  star  is  at  the  horizon,  £  is  90° ;  sin  £  =  1  and  cos  £  =  0; 

so  that  the  expression  becomes  cos  SS'M=  -     -  for  a  star  either 

cos  6 

rising  or  setting.     Substitute  this  in  (1)  and  we  have,  for  such  a 

star,   SM=  15'  cos  8  X ?  =  15'  sin  d>.    This  is  independent  of  the 

cos  8 

star's  declination,  is  therefore  the  same  for  all  stars  when  they  are 
rising  or  setting,  and  depends  only  on  the  latitude  of  the  observer. 

Since  this  motion  of  the  star  in  azimuth  is  merely  apparent,  and 
due  to  the  rotational  "skewing"  of  the  horizontal  plane  of  the 
observer,  the  demonstration  is  equivalent  to  that  given  in  Art.  141, 
in  connection  with  the  Foucault  pendulum. 

The  same  thing  may  of  course  be  proved  by  differentiating  the 

7  ^ 

equations  of  the  ZPS  triangle  so  as  to  find  the  value  of  —  in 

terms  of  £,  8,  and  <£,  making  £  =  90°  in  the  result ;  8  then  dis- 
appears as  above. 


584 


APPENDIX. 


1002,  Kepler's  Problem.  (Supplementary  to  Arts.  188,  189.) 
—  The  problem  is  to  find  the  place  which  a  body  will  occupy  at 
any  given  moment  when  moving  in  accordance  with  Kepler's  laws 
around  the  sun,  having  given  the  dimensions  of  its  orbit,  the  period 
of  revolution  T,  and  the  time  t  elapsed  since  the  planet  passed 
perihelion. 

In  Fig.  241  ABA'S'  is  the  orbital  ellipse,  having  AC  and  BC  as 
its  semiaxes,  respectively  designated  as  a  and  b.  S  is  the  focus, 

and  CS=ae,  e  being  the 
eccentricity  of  the  ellipse. 
P  is  the  position  of  the 
planet;  PS,  or  r,  is  the 
radius  vector,  and  the  angle 
ASP  is  the  true  anomaly 
designated  by  v.  The 
shaded  sector  ASP  is  the 
area  described  by  the  ra- 
dius vector  (in  accordance 
with  Kepler's  second  law), 
and  since  the  entire  area 
of  the  ellipse  is  Trab,  the 
area  of  the  sector  will  be 


FIG.  241.— Kepler's  Problem. 


irab-j  which  may  be  writ- 


ten %abX27r  —  '  The  second  factor  is  the  angle  (in  radians) 
found  by  multiplying  the  whole  circumference  by  the  fraction  of 
the  period  —  elapsed  since  perihelion,  and  is  known  as  the  Mean 

Anomaly,  being  the  anomaly  which  the  planet  would  have  if  its 
angular  motion  was  uniform.  It  is  usually  designated  by  M ; 

and,  if  expressed  in  degrees,  M=  360°—  • 

We  have  therefore 

Shaded  Sector  ASP  =  $abXM.  (1) 

Circumscribe  a  circle  around  the  ellipse,  through  P  draw  the 
ordinate  DPP',  and  join  CP  and  CP'.  The  angle  ACP'  is  called 
the  Eccentric  Anomaly,  and  is  usually  designated  by  u,  and  when 
determined  it  gives  the  means  of  easily  calculating  v  and  r. 


APPENDIX.  585 

The  shaded  sector  is  the  difference  between  the  elliptical  sector 
AGP  and  the  triangle  SCP. 

The  sector  ACP  is  less  than  the  circular  sector  ACP',  which 
equals  %a2u,  in  the  ratio  of  b  to  a,  since  from  the  properties  of 
the  ellipse  and  the  construction  of  the  figure  every  ordinate  in  the 
elliptical  sector  is  shorter  than  the  corresponding  ordinate  of  the 
circular  sector  in  the  ratio  of  PD  to  P'D,  or  of  BC  to  GC.  Hence, 
we  have 

Sector  ACP  =  -  X  i  a?u  =  %abu.  (2) 

In  the  triangle  SCP,  which  equals  £  PD  X  SC,  we  have 

PD  =  -  X  P'D  =  -  X  a  sin  u  =  b  sin  u : 
a  a 

and 

SC=ae. 

Therefore,  for  the  area  of  the  triangle,  we  have 

SCP  =  %abesinu.  (3) 

Subtracting  this  from  (2),  we  get 

ASP  =  %ab(u  —  e  sin  u).  (4) 

Comparing  this  with  (1),  we  have  the  equation  sought,  and  known 
as  Kepler's  equation,  viz., 

u  —  e  sin  u  =  M. 

This  is  a  "transcendental"  equation,  and  cannot  be  solved  by 
ordinary  trigonometrical  formulae,  but  the  value  of  u  can  easily  be 
found  by  approximation  when  M  is  given.  We  start  with  an 
approximate  value  for  u,  ascertain  how  nearly  it  satisfies  the  equa- 
tion, and  then  correct  the  assumed  value  so  as  to  diminish  the 
error,  repeating  the  process  until  a  satisfactory  value  is  obtained. 
There  are  also  various  formulae  which  give  the  value  of  u  in  rap- 
idly converging  series,  some  of  them  adapted  to  orbits  of  small 
eccentricity  like  those  of  the  planets,  and  others  to  orbits  nearly 
parabolic. 

When  u  is  found,  v  and  r  are  readily  determined  from  the 
triangle  PDS,  in  which  PD  =  b  sin  u,  and 

SD  (=  CD  —  CS)  =  a  cos  u  —  ae  =  a  (cos  u  —  e). 

Hence,  tan  v  = —  >  which   can   be   reduced   to   the  more 

a  cos  u  —  e 

Vji  G 
—  -  X  tan  £  u.      For  r  we  have 


586 


APPENDIX. 


Sea  e  if  o 


ad  an 


8         8 


APPENDIX.  587 

r  =  - —  =  b  -  — ;  or  we  may  use  the  equation  r  =  VPD2  -f-  &Z)2. 

sin  v         sin  v 

Substituting  the  values  of  PD  and  SD  given  above  and  reducing, 
we  get  r  =  a  —  ae  cos  u.  Still  again  we  may  use  the  general  equa- 
tion of  the  conic  (Art.  423),  r  =  H  ,  p 

1  +  e  cos  v 

1002*.  Graphical  Solution  for  u.  —  An  approximate  value  of  u 
is  easily  obtained  by  means  of  the  curve  of  sines,  as  shown  in  Fig. 
242.  If  from  any  point  c  on  the  curve  a  line  be  drawn  at  such  a 
slope  that  ad  -f-  cd  equals  the  eccentricity  of  the  orbit  in  question 
(i.e.  if  tan  acd  =  e),  then  ad  =  e  X  dc  =  e  sin  (c),  (c)  being  the  angle 
indicated  by  the  position  of  c  and  d  (82^-°  in  the  case  shown  in  the 
figure),  and  Ma  =  (c)  —  e  X  sin  (c),  i.e.  (c)  is  the  eccentric  anomaly 
corresponding  to  the  mean  anomaly  Ma  in  an  orbit  having  e  for 
its  eccentricity.  To  facilitate  the  construction  of  the  inclined  line 
at  the  proper  slope,  the  upper  horizontal  line,  marked  M-{-WQe, 
is  drawn  at  the  distance  of  100  units  from  the  base  line  M,  upon 
which  the  degrees  are  marked.  To  find  the  eccentric  anomaly  cor- 
responding to  a  given  mean  anomaly,  it  is  therefore  only  necessary 
to  mark  the  point  a,  corresponding  to  the  given  mean  anomaly 
(supposed  to  be  45°  in  the  figure),  and  on  the  upper  line  the  point 
b,  corresponding  to  M  + 100  e,  which  is  111.1  in  the  illustration, 
the  eccentricity  being  taken  as  0.661  from  the  example  given  in 
the  next  article.  Join  a  and  b,  and  the  point  c  gives  the  corre- 
sponding eccentric  anomaly.  (Generally  it  will  be  best  to  use  a  fine 
thread  to  join  a  and  b,  rather  than  to  trust  an  ordinary  ruler.)  Of 
course  the  joining  line  need  not  be  actually  drawn,  as  we  want 
only  its  point  of  intersection  with  the  curve.  If  the  mean  anomaly 
had  been  90°  instead  of  45°,  b  would  of  course  have  been  156.1, 
the  eccentricity  remaining  unchanged.  If  the  ellipse  had  an  eccen- 
tricity of  0.50  exactly,  then  for  a  mean  anomaly  of  45°  the  point 
on  the  upper  line  would  be  at  95  (b'  in  the  figure)  and  then  u 
would  be  at  c',  i.e.  72£°.  When  the  mean  anomaly  exceeds  100°, 
it  may  be  found  impossible  to  lay  off  M  +100e  as  directed.  In 
that  case  M  +  50  e  may  be  laid  off  on  the  "  fifty-line  ";  or,  if  M  is 
near  180°,  M  +  20  e,  laid  off  on  the  "  twenty-line,"  will  answer  the 
same  purpose. 

Owing  to  slight  imperfections  in  the  diagram,  due  to  inaccuracies 
in  the  drawing,  in  the  ruling  of  the  squares,  unequal  shrinkage  of 


588  APPENDIX. 

the  paper,  etc.,  the  values  of  u  obtained  from  it  are  only  approxi- 
mate, but  can  generally  be  relied  on  to  within  about  J-°.  This  is 
near  enough  for  many  purposes  (double-star  orbits,  for  instance), 
and  is  always  sufficient  as  the  starting-point  for  a  numerical  calcu- 
lation. 

1003.  Examples.  —  1.  Given  the  orbit  of  a  comet,  with  semi- 
major  axis  equal  to  four  astronomical  units,  and  eccentricity, 
0.66144  (or,  what  is  the  same  thing,  b  =  f  a,  as  drawn  in  Fig.  241). 
Required  the  place  of  the  comet  in  its  orbit  one  year  after  peri- 
helion passage.  Since  a  =  4,  the  period  =  4^  =  eight  years.  M  is 
therefore  one-eighth  of  the  circumference,  or  45°,  and  Kepler's 
equation  becomes  45°  =  u  —  0.66144  X  sin  u.  Since,  in  the  tables, 
sin  u  is  given  in  radians,  it  will  be  necessary  to  reduce  the  term 
containing  it  to  degrees  in  solving  the  equation :  this  may  be  done 

360° 
by  multiplying  the  term  by >  or  57°.2958,  which  gives  1.57861 

7T 

as  the  logarithm  of  the  coefficient  of  the  sinw.  We  then  have 
45°  =  u°  —  [1.57861]°  sin  u  as  the  equation  to  be  solved. 

We  get  the  first  approximation  from  the  sine-curve,  as  shown  in 
the  preceding  article,  and  find  %  =  82°  30' ;  we  then  proceed  to 

test  it  as  follows  : 

sin  82°  30' 9.99627 

log  of  coeff.   .  .  .  1.67861 

37°.  573 1.57488 

82°.50Q 
Difference  44°.927   (instead  of  45°). 

This  value  of  u  is  not  quite  large  enough  and  must  be  increased  : 
it  will  be  noticed  that,  as  the  angle  is  very  near  90°,  its  sine  will 
change  very  slowly,  and  the  term,  [1.57861]°  sinw,  will  be  only 
very  slightly  altered  by  any  small  change  in  u.  We  must  there- 
fore increase  u'  by  only  a  very  little  more  than  the  difference, 
0°.073,  between  44°.927  and  45°.  We  assume  accordingly  for  a 
second  approximation  uz=  82°.58  or  82°  34 '.8.  We  then  have  : 

sin  82°  34'.  8 9.99635 

coeff 1.57861 

37°.680 1.57496 

82°.  580 
Difference  45°.  000, 

and  this  satisfies  the  equation  exactly,  so  far  as  can  be  determined 
with  five-place  logarithms.  When  great  precision  is  required,  it  is 


APPENDIX.  589 

necessary,  of  course,  to  use  seven  places.     In  this  example  the 
exact  value,  so  computed,  is  82°.58042  =  82°  34'  49".5. 
To  get  v,  we  use  the  formula  for  -J-  v. 

l+e=  1.66144  log  0.22049 
1  —  e  -  0.33856  9.52962 
2)0.69087 

log  0.34543 

iu  =  41°  17'.4  tang          9.94360 

tang^r     0.28903 
i  v  =  62°  47'. 8.         Hence,  v  =  125°  35'. 6. 

For  r,  we  use  the  formula  r  =  a  —  ae  cos  u. 

a  =  4  log  0.60206 

e  =  0.66144  log  9.82049 
cos  82°  34'.  8  9.11106 

ae  cos  M  =  0.3417  9.53361 

a  =  4.0000 

r  =  3.6583. 

2.  In  the  same  orbit,  let  t  =  2  years.     Find  u,  v,  and  r. 

Ans.  u  =  122°  06'.  2 
0<=161°6r.7 

r  =  5.4061. 

3.  Let  £  =  3  years  in  same  orbit. 

Ans.  u  =  152°30/.0 
v  =  167°  23'.6 
r  =  6.3468. 

1004.  Projection  of  a  Lunar  Eclipse.  (Supplementary  to  Art. 
878.)  —  We  take  as  an  example  the  eclipse  of  Sept.  3,  1895,  for 
which  we  find  the  following  data  in  the  "  American  Ephemeris," 
p.  414  : 

Greenwich  mean  time  of  opposition  in  right  ascension,  Sept.  3,  17h  47m  468.6. 


Sun's  decimation  +  7°  17'  2".  5 

"  hourly  motion  in  decl.  —  55".  3 
"  "  "  "  E.A.  98.04 
"  semi-diameter  15'  52".  1 

"      horizontal  parallax  8".  5 


Moon's  declination  —  7°  25'  54//.8 
"  hourly  motion  in  decl.  +  13'  44".  6 
«*  "  «'  "  R.A.  105".82 

"  semi-diameter  14'  41".  8 

"  horizontal  parallax         63'  58".  4 


A  convenient  scale  is  1000"  to  the  inch:  this  will  bring  the 
diagram  within  the  limits  of  an  8  by  10  sheet  of  paper  and  is 
large  enough  to  give  all  the  accuracy  that  is  required.  Fractions 
of  a  second  are  of  course  neglected. 


590 


APPENDIX. 


I.  The  first  step  is  to  lay  off  the  "  relative  orbit "  of  the  moon 
with  respect  to  the  shadow.  Draw  two  lines  accurately  perpen- 
dicular to  each  other,  their  point  of  crossing  0  (Fig.  243)  being 
the  position  of  the  moon's  centre  at  the  given  moment  of  opposition. 

(a)  On  the  horizontal  line  EW  lay  off  the  difference  of  the 
hourly  motions  of  the  sun  and  moon  in  right  ascension,  in  seconds 


FIG.  243. 

of  arc,  reduced  to  seconds  of  a  great  circle  by  multiplying  by  the 
cosine  of  the  moon's  declination.     In  this  case  we  have 
(105.82  —  9.04)  X  15  X  cos  7°  25'.9 

=  96.78  X  15  X  cos  7°  25'.9  =  1439".5. 

Ob  and  Od  are  each  laid  off  with  this  value,  while  Oa  and  Oe  are 
twice  as  great. 

(b)  At  b  and  d  lay  off  the  difference  of  the  hourly  motions  of 
declination,  remembering  that  the  centre  of  the  shadow  moves 
north  when  the  sun  moves  south.     We  have  in  this  case 
13'  44".6  —  55".3  =  824".6  -  55".3  =  769".3. 

(This  requires  no  reduction,  since  declination  is  measured  on  the 
hour-circle.)  We  lay  off  there  the  ordinates  at  b  and  d,  each  equal 
to  769.3,  and  those  at  a  and  e  are  twice  as  great.  Since  the  moon 
is  moving  northwards,  the  ordinates  to  the  west  (right)  of  0  are 
laid  off  downwards,  and  those  on  the  east  side  upwards. 


APPENDIX.  591 

(c)  If  the  work  has  been  properly  done,  the  four  points  thus 
determined  will  all  lie  precisely  on  a  straight  line  passing  through 
0,  and  will  be  the  points  occupied  by  the  moon's  centre,  exactly 
one  hour  and  two  hours  before  and  after  the  moment  of  opposition. 
The  line  is  to  be  divided  to  mark  the  half  and  quarter  hours,  and 
when  afterwards  found  necessary,  the  fifteen-minute  spaces  can  be 
divided  into  three  five-minute  spaces.  , 

II.  Mark  the  centre  of  the  shadow.  Lay  off  a  distance  OC  north 
or  south  of  0,  equal  to  the  difference  between  the  declinations  of 
the  sun  and  moon  ;  in  this  case 

(7°  25'  54".8  —  7°  17'  2".5)  =  8'  52".3  =  532".3. 
It  is  laid  off  to  the  north  of  0  because  the  centre  of  the  shadow 
(opposite  the  sun)  has  a  smaller  southern  declination  than  the 
moon. 


III.  Draw  the  shadow.     Its  radius  (Art.  372)  is  (P+p  — 
or,  in  this  case, 

ft  (53'  58".4  +  8".5  -  15'  52».1)=  f  J  (2294  ".8)  =  2333". 

With  C  as  a  centre  and  this  radius,  describe  the  large  circle  which 
represents  the  shadow.  (It  is  not  necessary  actually  to  draw  the 
shadow  circle,  but  it  is  usual  to  do  so  :  the  radius  of  the  shadow  is 
needed  for  the  next  step.) 

IV.  Mark  the  points  on  the  relative  orbit  occupied  by  the  moon's 
centre  at  the  moments  of  contact  with  the  shadow,     (a)  To  the  radius 
of  the  shadow  add  the  semi-diameter  of  the  moon  (in  this  case, 
2333"  +  882"  =  3215"),  and  with  this  distance  as  a  radius,  from  the 
centre  C  strike  two  arcs  cutting  the  relative  orbit  at  I  and  IV, 
which  will  be  the  position  of  the  moon's  centre  at  the  first  and  last 
(external)  contacts. 

(b)  Subtract  the  moon's  semi-diameter  from  the  radius  of  the 
shadow,  and  with  this  difference  (2333"—  882"  =  1451")  as  a 
radius  from  C  find  the  points  II  and  III,  of  internal  contact.  The 
figure  may  be  completed  by  drawing  the  circles  to  represent  the 
moon  (radius  882"),  using  I,  II,  M9  III,  and  IV  as  centres.  But  it 
is  not  necessary.  M}  the  middle  of  the  eclipse,  is  half-way  between 
II  and  III. 

V.  Finally,  read  off  the  times  of  the  contact  on  the  relative  orbit 
regarded  as  a  scale  of  time.     For  contacts  I  and  II  subtract  the 


592  APPENDIX. 

reading  from  the  time  of  opposition  (17h  47m.8  in  this  case) ;  for  III 
and  IV,  add.     In  the  present  case  the  results  come  out  as  follows : 


I. 

h.  m. 

-  1  47.5 

17  47.8 

II. 

h.  m. 

-0  41.5 
17  47.8 

MIDDLE. 

h.  m. 
+  0   9.5 

17  47.8 

III. 

h.  m. 
+  1  00.0 

17  47.8 

IV. 

h.  m. 

+  2  7.0 
17  47.8 

16  00.3 
(15  59.9)  G.M.T. 

17  06.3 
(17  06.4) 

17  57.3 
(17  57.0) 

18  47.8 
(18  47.5) 

19  54.8 
(19  53.9) 

The  figures  in  parentheses  are  the  calculated  results  given  in  the  almanac. 
To  get  Eastern  Standard  time,  subtract  5  hours. 

1005.    Calculation  of  a  Lunar  Eclipse,  —  I.   Preparation  of  data. 

(a)  Moon's  motion  in  R.A.  105s.  82 

Sun's        "        «»     ««  9fl.Q4 

,       96  .78 
X  15  (add  half  and  multiply  by  10)      48.39 

1451".  7  log  3.16188 

cos  7°  25'.  9  9.99633 

O6=1439".5  3.15821 

(6)  bt  =  13'  44".6  -  55".3  =  12'  49".3  =  769".3 

(c)  OC  =  7°  25'  54".8  -  7°  17'  2".5  =  8'  52".3  =  532".3 

(d)  Radius  of  shadow  =  (P  +  p  -  S)  P  53'  58".4  3 

p          8".5  - 

54'  06".  9 

S  15'  52".  1 

38'  14".  8 

add  &  38".2 

r  =  38'  53".  0  =  2333".0 

(e)  Semi-diameter  of  moon  =  s  =  14'  41".  8  =  881".  8 

(  p  =  C,I  =  (r  +  s)  =  3214-.8 
^  J  (  p'=C,  II  =-(r  -  s)  =  1451".2 
(g)  Time  of  opposition  =  17h  47m  468.6  =  17h  47I».78 

II.    Calculation  of  relative  orbit  (triangle  Obt)  to  find  angle  bOt 
(i),  and  hourly  orbital  motion  Ot. 

bt  769".3  log  2.88610 

—  u^,,^  3.15821 


i  =  28°07/3  tang  9.  72789 

YTA  nt  =  -™_  3-16821 

cos  i  cos  28°  07'.3  9.94544 

Hourly  motion  (only  the  logarithm  is  needed)  3.21277 

III.    Calculation  of  CM  and  OM  (triangle  OCM). 

(a)  CM=  OCcosi  532".3  log  2.  72616 

cosi  9.94544 

CM  =  469".6  2.67160 


APPENDIX.  593 

(6)  OM  =  OCsmi  532".3  2.72616 

sini  9.67335 

Log  OM  (seconds  of  arc)  2.39951 

To  reduce  to  time,  divide  by  hourly  motion  3.21277 

OJf(time)=    0*.1537  9.18674 

=    0*09m.22 
Time  of  opposition  =  17*  47m.78 

17*  57m.OO  =  middle  of  eclipse. 

IV.  Calculation  of  M,  I  (=  M,  IV),  and  angle  77,  and  of  time  of 
first  and  last  contacts  (triangle  MCT). 

_CM_  469".6  2.67160 

n77~cTl~3214^8  3.50715 

77  =  8°  23'.  8  sin  77                        9.16445 

M,  I  =  O,  I  cos  77  3.50715 

cos  8°  23'.8  9.99533 

(b)  Log  Jf,  I  (seconds  of  arc)  3.50248 

Divide  by  hourly  motion  3.21277 

Jf,  I  (time)  =  1*.9485  =  1*  56m.91  0.28971 

Middle            17*  57m.OO  17*  57ra.OO 

-  1*  56m.91  +  1*  56m.91 

(I)     16*00m.09  19*53m.91     (IV) 

V.  Calculation  of  M,  II  (=  M,  III)  and  of  angle  0,  and  of  time 
of  the  two  internal  contacts  (triangle  MCIT). 

c-    0-  CM  -  469//- 6  2.67160 

~  C,  II  ~  1451".2  3.16173 

0  =  18°  52'.  5  sin0                       9.50987 

(6)  Jf,  II  =  (7,  II  X  cos  8  cos  18°  62'.  5          9.97600- 

3.16173 

M,  II  (seconds  of  arc)  3.13773 

Divide  by  orbital  hourly  motion  3.21277 

M,  II  (time)  =  0*.841  =  50m.46  9.92496 

Middle            17*  57m.OO  17*  57m.OO 

—  50m.46  4-  50m.46 

(II)     17*  06m.54  18*  47m.46     (III) 

VI.  The  angles  -q  and  0  determine  the  arcs  of  the  moon's  limb 
intercepted  between  the  north  point  of  the  limb  and  the  point  of 
contact. 

For  the  1st  contact  the  arc  71^  =  (90°  —  <*)—  17  =  53°  29';  for 
the  2d,  the  arc  nzk2  =  (90°  +  i)  +  0  =  137° ;  for  the  3d  contact, 
nsks  =  (90°  -  i)  +  6  =  80°  45' ;  for  the  4th  we  have  n^  =  (90°  +  tj 
—  rj  =  109°  44'.  The  1st  and.  3d  are  reckoned  from  the  north 
towards  the  east,  the  2d  and  4th  towards  the  west. 


594 


APPENDIX. 


Attention  is  called  to  the  fact  that  the  assumption  of  a  uniform 
motion  of  the  moon  during  the  four  hours  involved  in  the  calcula- 
tion is  not  correct  and  would  not  be  permissible  if  the  phenomena 
of  the  eclipse  could  be  precisely  observed.  Since,  however,  it  is 
impossible  to  be  sure  of  the  tenths  or  even  quarters  of  a  minute  in 
observation,  the  method  of  calculation  is  abundantly  accurate  for 
its  purpose. 

1006.  Proof  that  the  Orbit  described  under  the  Law  of  Gravita- 
tion is  a  Focal  Conic.  (Supplementary  to  Arts.  421,  424.)  —  (The 
demonstration  that  follows  is  substantially  the  same  as  one  given 
in  Williamson's  "  Treatise  011  Dynamics.7') 

1.  General  differential  equations  of  the  motion.  Let  the  particle 
P  (Fig.  244)  be  urged  towards  0  along  r  by  a  force  F,  making  the 

angle  6  with  OX,  the  axis  of 
X.  The  force  can  be  resolved 
into  two  components  along  the 
axes  of  X  and  Y,  viz.,  F  cos  0 

and  F  sin  0,  or  F-  and  F^- 
r  r 

Hence,  the  accelerations  along 
the  axes  will  be  given  by  the 


d2X  X 

equations     -^  =  —  F  -  > 


dt 


and 


FIG.  244 


dt? 


==•  —  F—  j  the  minus  sign 
being  used  because  the  force  F  tends  to  diminish  both  x  and  y. 
If  now,  according  to  the  law  of  gravitation,  F=  ^>  the  equations 

become  -r-^-  =  —  ^-  and  — ^  =  •—£-£  (1).     The  integration  of  these 
equations  will  give  the  law  of  motion  and  the  nature  of  the  orbit. 

2.    Equable  description  of  areas.     Multiply  the  two  equations  (1) 
by  y  and  x,  respectively,  and  subtract  the  first  product  from  the 

second  :  we  get  x-^  —  y -|  =  0,  or  — (  x-j-  —  y-jf-  }  =  0.      Hence, 
at  air  at\   at          at  J 

dy         dx 
by  integrating,  x-£  —  y  —  =  h,      (2);    h    being    the    constant    of 

duL  CliC 

integration  and  independent  of  the  time. 


APPENDIX.  595 

If  in  (2)  we  substitute  r  cos  0  for  x,  and  r  sin  0  for  y,  and  perform 
the  necessary  reductions,  we  get  the  corresponding  polar  differen- 

tial equation  r*—  =  h  (3). 

The  left-hand  members  of  both  (2)  and  (3)  are  the  well-known 
expressions  for  twice  the  increment  of  the  area  of  the  sector  of  the 
curve,  corresponding  to  consecutive  values  of  x  and  y,  or  of  r  and  6  ; 
so  that  the  equations  prove  that  this  increment  of  area  in  a  unit  of 
time  is  constant,  and  constitute  an  analytical  demonstration  of  the. 
principle  proved  geometrically  in  Arts.  402-405. 

From  (3)  we  have  also  ~$  —  j)~ji  (4)- 

3.    Nature  of  the  orbit.     Substitute  in  equations  (1)  this  value 

,1  L    d?x  fjuxde  p         .dO 

of  -=>   from   (4),  and  we  get   -73  =  —  T~~JI  =  ~^  cos  ^31*  and 
ir  di  fi  r  at  h,  at 


Integrating,  we  have  -J7=—  v  sin  0+a,  and  -^=-f-^cos04-/3  (6), 
at          ft  dt          hi 

a  and  ft  being  the  constants  of  integration,  depending  upon  the 
initial  conditions  of  motion  at  P. 

4.    Substitute  these  values  of  -^  and  ~  in  equation  (2),  and  we 


have 


j-(  x  cos  0  +  y  sin  6  J 


or  j-(  x  cos  0  +  y  sin  0  J  +  fix  -f-  ay  —  h  =  0. 

But  (x  cosO  +  y  sin  0)  =  Om  +  mP  (Fig.  244)  =  r,  so  that  finally 
this  equation  of  the  gravitational  orbit  becomes 


This  is  a  form  of  the  general  equation  of  a  conic,  with  the  origin 
at  the  focus.  The  coefficients  of  x  and  y  depend  upon  the  eccen- 
tricity of  the  curve,  and  the  angle  between  its  major  axis  and  the 
axis  of  abscissas,  while  the  absolute  term  is  the  semi-parameter, 

usually  designated  by  p,  or  a  (1  —  e2). 

fit 
From  (7)  we  see  that  in  the  gravitational  orbit  p  =  —  »  or  h  =  V^p  (8)  ;  and 

since  n  in  the  solar  system  is  the  mass  of  the  swn,  we  have  h  proportional  to  the 


596 


APPENDIX. 


square  root  of  the  parameter  of  the  orbit,  in  accordance  with  the  form  of 
Kepler's  third  law  given  at  the  end  of  Art.  423. 

5.    Transformation  of  equation  (7)  to  the  normal  polar  form.     Put 
—  =p,  and  make  the  coefficients  of  x  and  y,  respectively,  equal  to 

e  cos  \ff  and  e  sin  \j/  :    also  for  x  write  r  cos  0,  and  for  y,  r  sin  0, 
9  being  the  vectorial  angle.     The  equation  becomes 

r  +  e  cos  \l/  X  r  cos  0  -\-  e  sin  ^  X  r  sin  0  —  ^?  =  0, 
or  r  (1  +  e  cos  0  cos  ^  +  e  sin  0  sin  \l/)  =p, 

or  r  (1  +  e  cos  [0  —  ^])  =  jp, 


or,  finally, 


which  is  of  the  normal  form,  e  being  the  eccentricity  of  the  conic, 
while  the  major  axis  of  the  conic  makes  the  angle  \j/  with  the 
axis  of  abscissas,  so  that  (0  —  \l/)  is  the  anomaly,  v,  in  the  equa- 

tion as  usually  written,  r  =  .    ,      -- 
J  1  +  e  cos  v 


1007.    Expression  for  the  Velocity  at  any  Point  of  the  Orbit. 
(Supplementary  to  Art.  428.)  —  Suppose  the  orbit  elliptical,  and 

that  the  shaded  sector  in 
Fig.  245  has  been  described 
in  a  unit  of  time  (say  one 
second);  then  the  area  of 
this  sector  is  the  planet's 

areal-velocity    —  >    and   the 

short  base  of  the  sector  (at 
P)  is  its  linear  velocity  V. 
At  P  draw  the  tangent 

MPN,  and  from  the  two  foci  S  and  F  draw  the  two  perpendiculars 
to  it, /and  f.    Also  join  PF,  or  rf.     The  area  of  the  sector  equals 

V  X  f      h  h? 

— — £  =  -,  so  that  V2  =  -^-     But  by  equation  (8)  of  the  preceding 

«/ 

article  h2  =  pp  ;  and,  by  the  properties  of  the  ellipse,  p  =  —     We 
have,  therefore,  V2  =  ^-  • 


APPENDIX.  597 

f        1* 
Again,    from   the   properties   of   the   ellipse,  fff  =  b2  ;    —  =  — 

(similar  triangles)  ;  and  (r  +  r')  =  2  a. 


Hence,    F*= 

af      a  r       a 


finally 


,  V2  =  p(  ----  J,  as  given  in  Art.  428. 


A  corresponding  demonstration  holds  for  the  hyperbola,  bearing 
in  mind  that  in  that  conic  a  and  &2  are  both  negative.     The  final 

equation  is  the  same  :  in  either  case  —  =  —  —  —  F2,  and  a  is  posi- 

a         r 

2  u, 
tive  (ellipse)  when  F2  <  -—  ;  becomes  negative  (hyperbola)  when 

F>  —  ;  and  is  infinite  (parabola)  when  F2=  —  • 


1008.  Proof  that  the  "  Parabolic  Velocity,"   V,  equals 

(Supplementary  to  Art.  429.)  —  When  the  attracting  force  =  —j  we 

have    for   the    acceleration    of   fall   towards   the    attracting   body 

dv  a      _.  dr  .  .   .    .         vdv  f  '  p\dr 

—  =  —£•     But  v  =  —  -     Multiplying,  —  =  -  Ifiljjf     Integrat- 
ing we  get,  —  =  -  +  C.     If  w  =  0  when  r  =  s,  then  (7  =  —  -;  whence 

we  have,  v2  =  2/x  f  -       -  )  •    If  we  make  s  infinite  in  this  expression, 

V'      s  / 
v  becomes  the  "velocity  from  infinity,"  or  "parabolic  velocity," 

denoted  by  U;  and  we  have,  therefore,  Z72=— ,  or  V=*\ — • 

1009.  (Supplementary  to  Art.  493.) — Fig.  246  shows  how  the 
combination  of  the  earth's  motion  with  that  of  a  planet  produces 
an  epicycloid  as  the  relative  (apparent)  path  of  the  planet.     The 
earth's  orbit  is  represented  by  the  smaller  circle,  upon  which  are 
marked  eight  points,  0-VII,  occupied  at  eight  equidistant  times. 
AB  is  part  of  the  circular  orbit  of  a  second  planet  with  a  period  of 
twelve  years  (nearly  the  case  of  Jupiter),  the  points  marked  01?  02, 
and  03  being  those  occupied  by  the  planet  when  the  earth  is  at  O. 


598 


APPENDIX. 


In  the  same  way  the  points  lj,  12,  and  13  correspond  to  I  in  the 
earth's  orbit,  and  similarly  for  the  other  points  marked  2,  3,  etc. 
At  the  start  the  earth  is  supposed  to  be  at  0,  and  the  planet  at  Olt 
Around  Q1  as  a  centre  draw  a  circle  equal  to  the  earth's  orbit,  and 
draw  the  radius  OjO'  parallel  to  SO.  Then  the  line  SO'  will  be 
parallel  and  equal  to  00l5  so  that  Oj  has  the  same  distance  and 
direction  from  S  as  Ox  has  from  0:  in  other  words,  if  the  earth 


FIG.  246. 

were  transferred  to  S,  0'  would  occupy  precisely  the  same  position 
in  the  celestial  sphere  that  the  planet  actually  does  as  seen  from  0. 
In  an  eighth  of  a  year  the  earth  will  have  moved  to  I,  and  the 
planet  to  1^ :  the  line  Ilx  will  represent  the  new  direction  and 
distance  of  the  planet,  and,  repeating  the  same  construction  as 
before  (i.e.  drawing  from  lj  a  line  lxl'  parallel  and  equal  to  the 
radius  ST),  we  find  V  as  the  point  which,  seen  from  S,  would  hold 
the  same  relative  position  that  l^  does  with  respect  to  I.  Now  a 


APPENDIX.  599 

glance  at  the  figure  shows  that  this  point  1'  is  on  the  circumference 
of  a  circle  precisely  like  the  one  first  drawn,  but  with  its  centre 
moved  to  11?  and  is  at  the  extremity  of  a  radius  inclined  45°  to  the 
original  radius  (^O',  while  the  curved  line  O'l1  is  the  line  along 
which  the  planet  would  have  appeared  to  move,  if  the  earth  were 
regarded  as  stationary  at  S.  Carrying  out  the  construction  for  the 
successive  positions  of  the  earth  and  the  planet,  we  find  the  points 
2',  3',  4',  etc.,  for  the  apparent  "  geocentric  "  positions  of  the  planet, 
and  the  looped  curve  is  its  apparent  geocentric  path.  The  points 
P,  where  the  planet's  distance  from  the  earth  is  least,  and  the 
apparent  motion  retrograde,  correspond  to  opposition,  when  planet 
and  earth  are  on  the  same  side  of  the  sun.  The  points  of  maxi- 
mum distance,  marked  J,  are  those  of  conjunction,  when  the  earth 
and  planet  are  on  opposite  sides. 


THE   GREEK   ALPHABET. 


Letters. 

Name. 

Letters. 

Name. 

Letters. 

Name. 

A,  a, 

Alpha. 

I,  t, 

Iota. 

P,  p  g, 

Rho. 

B,  j8, 

Beta. 

K,   K, 

Kappa. 

5,   O*  9, 

Sigma. 

r,  y, 

Gamma. 

A,  X, 

Lambda. 

T,  r, 

Tau. 

A,  8, 

Delta. 

M,  /*, 

Mu. 

Y,  v, 

Upsilon. 

E,  e, 

Epsilon. 

N,  v, 

Nu. 

^?  </>» 

Phi. 

z?  £, 

Zeta. 

H,  & 

Xi. 

x'  X' 

CM. 

H,  >/, 

Eta. 

0,  o, 

O  micron. 

*»  «A» 

Psi. 

6  #,   Theta.  H,  TT  o,    Pi.  O,  w,      Omega. 


MISCELLANEOUS   SYMBOLS. 

6  ,  Conjunction.  A.R.,  or  a,  Right  Ascension. 

D  ,  Quadrature.  Decl.,  or  8,  Declination. 

<?  ,  Opposition.  X,  Longitude  (Celestial). 

&,  Ascending  Node.  /:?,  Latitude  (Celestial^. 

^5,  Descending  Node.  <£,  Latitude  (Terrestrial), 

o),  Angle  between  line  of  nodes  and  line  of  apsides.     Also 
obliquity  of  the  ecliptic. 


DIMENSIONS    OF   THE   TERRESTRIAL   SPHEROID. 

(According  to  Clarke's  Spheroid  of  1878.     For  the  spheroid  of  1866,  see  Art.  145.) 

Equatorial  semidiameter,  — 

20  926  202  feet  =  3963.296  miles  =  6  378  190  metres. 

Polar  semidiameter,  — 

20  854  895  feet  =  3949.790  miles  =  6  356  456  metres. 

Oblateness  (Clarke),  g  9  ^  4  6  ;    (Harkness),  _1_. 


Length  (in  metres)  of  1°  of  meridian  in  lat.  <£  =  111  132.09  —  556.05 
cos2<£  +  1.20  cos 4<£. 

Length  (in  metres)  of  1°  of  parallel,  in  lat.  <f>  =  111  415.10  cos<£  — 
94.54  cos  3  6. 


602  APPENDIX. 

1°  of  lat.  at       pole  =  111  699.3  metres  =  69.407  miles. 
1°  of  lat.  at  equator  =  110  567.2  metres  =  68.704  miles. 

These  formulae  correspond  to  the  Clarke  Spheroid  of  1866,  used  by 
the  U.S.  Coast  and  Geodetic  Survey. 


TIME   CONSTANTS. 

The  sidereal  day        =  23h  56m  48.090  of  mean  solar  time. 
The  mean  solar  day  =  24h  3m  568.556  of  sidereal  time. 

To  reduce  a  time-interval  expressed  in  units  of  mean  solar  time  to 
units  of  sidereal  time,  multiply  by  1.00273791  ;  Log.  of  0.00273791 
=  [7.4374191]. 

To  reduce  a  time-interval  expressed  in  units  of  sidereal  time  to 
units  of  mean  solar  time,  multiply  by  0.99726957  =  (1  —  0.00273043)  ; 
Log.  0.00273043=  [7.4362316]. 

Tropical  year  (Newcomb,  reduced  to  1900),  365d  5h  48m  458.98. 
Sidereal  year  "  "  "       365    6      9        8.97. 

Anomalistic  year    "  "  "       365    6    13      48.09. 

Mean  synodical  month  (Nelson),  29d  12h  44m  28.864. 

Sidereal  month, 27      7    43    11.545. 

Tropical  month  (equinox  to  equinox),  .27  7  43  4.68. 
Anomalistic  month  (perigee  to  perigee),  .  27  13  18  37.44. 
Nodical  month  (node  to  node),  .  .  27  5  5  35.81. 


Obliquity  of  the  ecliptic  (Newcomb), 

23°  27'  8".26  —  0".468  (t  — 1900). 

Constant  of  precession  (Newcomb),  50".248  +  0.000222  (t  — 1900). 
Constant  of  nutation  (Paris  Conference,  1896),         9".21. 
Constant  of  aberration  (Paris  Conference,  1896),    20".47. 
Solar  parallax  (Paris  Conference,  1896),  8".80. 

Velocity  of  light  (Michelson  and  Newcomb), 

186330  miles,  299860  km. 


APPENDIX. 


603 


B  81 


Major 
Planets. 


ITer'striall 
|  Planets.  | 


e 


oo°° 


$3*8 


opoo 


- 


0000 

gis 


r 


nO    SYMBOL. 


8  n 


4-  -i  -i  ;.; 


B 





P    0000 


SB 


15'  (to 
6°  33' 


Apparen 
Angular 
Diamete 


Axia 
Rotati 


Inclinat 
of  Equa 
to  Orbi 


on 
or 


Oblate- 
ness. 


Albedo 
(Miiller). 


. 


Major 
Planets. 


•S-       c. 

g  g  3  | 


b>  so 


oo   oo   o< 

r1  .»  .« 


^4    O     t-    00 

00    U-    05    03 


!;1  ill 


N|S    S 


B 

00^ 


Terrestrial 
Planets. 


Mercury. 
Venus  .  . 
The  Earth 
Mars  ... 


M    M    0    0 


P    *    P 

CO    io    O 


l->    M    O    O 
gg    §    g    g 


P1  ?°  i-1 


MOCO    -g 

S  8  8  S 

to  o  «<  oo 


i^.  -j 

oo  to  o> 

8  S3  S 

S&g 


IS? 

52  ^  - 


SYMBOL. 


sfe 


Sid 
Pe 
(mea 
da 


1 


nclinat 
to 
Eclipti 


gitude  o 
scending 
Node. 


Longitude 
of 
Perihelion. 


gii 

-  * 


604 


APPENDIX. 


TABLE  II.— THE   SATELLITES 


NAME. 

Discovery. 

Dist.  in  Equa- 
torial Radii 
of  Planet. 

Mean 
Distance  in 
Miles. 

Sidereal  Period. 

60.27035 

238840 

27d    7h  43m  11"  5 

SATELLITES  OF 


1 

2 

Phobos    .... 
Deimos    .... 

Hall,                       1877 

2.771 
6.921 

5850 
14650 

7fc  39™  15».l 
I*     6   17    54.0 

SATELLITES  OF 


5 

Nameless    .     .    . 

Barnard,                1892 

2.551 

112500 

llh  57"  22*.6 

1 

lo 

Galileo,                  1610 

5.933 

261  000 

Id  18   27    33  5 

2 

Europa   .... 

9.439 

415000 

3    13    13    42.1 

3 

Ganymede  .     .    . 

«                            « 

15.057 

664000 

7      3   42    33.4 

4 

Callisto  .... 

"                            " 

26.486 

1167000 

16    16  32    11.2 

6 

Nameless     .    .    . 

Perrine,                 1905 

162.92 

7185000 

253.4 

7 

Nameless     .    .    . 

"                            " 

167.86 

7403000 

265.0 

SATELLITES  OF 


1 

Mimas     ....... 

W.  Herschel,        1789 

3.11 

117000 

22h37»    5*.7 

2 

Enceladus   .     .    . 

«           « 

3.98 

157000 

11     8   53      6.9 

3 

Tethys     .... 

J.  D.  Cassini,        1684 

4.95 

186000 

1    21    18    25.6 

4 

Dione 

«         «                   a 

6.34 

238000 

2    17   41      9.3 

5 

Rhea  .    . 

"          "                 1672 

8.86 

332000 

4    12   25    11.6 

6 

Titan 

Huyghens,             1655 

20.48 

771  000 

15    22   41    23.2 

7 

Hyperion     .    .     . 

G.  P.  Bond,            1848 

25.07 

934000 

21      6   39    27.0 

8 

lapetus   .... 

J.  D.  Cassini,         1671 

59.58 

2225000 

79      7   54    17.1 

9 

Phoebe    .... 

W.  Pickering,       1898 

213.5 

8000000 

546.5 

10 

Themis    .... 

"           "                1905 

24.3?" 

906000? 

20    20 

SATELLITES  OF 


1 

Ariel  

Lassell,                  1851 

7.52 

120000 

2*  12»>  29°>  2K1 

2 

Umbriel  .... 

10.46 

167000 

4      3   27    37.2 

3 

Titania    .... 

W.  Herschel,        1787 

17.12 

273000 

8    16   56    295 

4 

Oberon    .... 

«                 (C                                     « 

22.90 

365000 

13    11     7      6.4 

SATELLITE  OF 


1 

Nameless     .    .    . 

Lassell,                  1846 

12.93 

221500 

5d  21h    2"  44».2 

APPENDIX. 


605 


OF  THE  SOLAR  SYSTEM. 


Synodic  Period. 

Inc.  of  Orbit 
to  Ecliptic 

Inc.  to  Plane  of 
Planet's  Orbit. 

Eccen- 
tricity. 

Diam'r 
in  Miles 

Mass  in 
Terms  of 
Primary. 

Remarks. 

5°    08'    40" 

•  "-  ':'-  ' 

0.05491 

2162 

i 

81.6 

Specific  gravity 

MARS. 


26°    17'.2 

28°  ± 

0 

35? 

? 

Orbits  sensibly 

coincident    with 

- 

25     47.2 

28°  ± 

0 

10? 

? 

planet's  equator. 

JUPITER. 


2°    20'    23" 

_ 

? 

100? 

? 

Id  ish  28»  35*.9 

2     08       3 

_ 

0 

2500 

.00001688 

3   13  17    53.7 
7     3   59    35.9 

1      38     57 
1      59     53 

- 

0 
.0013 

2100 
3550 

.00002323 
.00008844 

The  diameters 
are  Engelmann's. 
The  rest  of  the 

16   18     5      6.9 

1      57     00 

_ 

.0072 

2960 

.00004248 

data  are  from 
Damoiseau. 

28°  4  (  to  plane  of 

100? 

,10.J    planet's 

rf1  •*  (    equator 

40? 

SATURN. 


Long,  of  Ascend. 
Node  of  orbits 
on  ecliptic  for 
1900,  168°  10'  35". 

28°    10'    10" 

About  27°. 
Inclination  of  the 
5  inner  satellites 
to  plane  of  celes- 

0 
0 
0 
0 

600? 
800? 
1200? 
1100? 

9 

The  planes  of 
the  5  inner  orbits 
sensibly  coincide 
with  the  plane  of 
the  ring. 

(5  inner  satellites 

" 

tial  equator 

0 

1500? 

? 

and  ring.). 

27      38     49 

27       4.8 

=  6°  57'  43"  (1900) 

.0299 
.1189 

3500? 
500? 

38*00 

(     Discovered  in- 
j  dependently  by 

' 

18     31.5 
5       6 
39      00? 

-      . 

.0296 

.22 
.23 

2000? 
50? 
30? 

; 

(  On  photographs. 
(Retrograde. 
On  photographs. 

URANUS. 


Long,  of  Ascend. 

97°    51' 

0 

500? 

9 

—     g2°  Q9' 

Node  of  orbits  on 

Inc.  to  celestial 

0 

400? 

? 

plane  of  ecliptic 

« 

equator  75°  18' 

0 

1000? 

All  Retrograde. 

=  165°  32'  (1900). 

«            « 

(1900). 

0 

800? 

? 

NEPTUNE. 


Long.  Asc.  Node,  145°  12' 
184°  25'  (1900).                 =-34°  48' 

120°  05'  (1900) 

0 

2000? 

9 

Retrograde. 

606 


APPENDIX. 


31 

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Longitude  of 
Asc.  Node. 

GO  IO 
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00  b- 

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TH  0 
3  2 

rH 
0 
rH 

SCO 
CO 

CM  CM 

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APPENDIX. 


607 


TABLE   IV.— STELLAR   PARALLAXES  AND   PROPER   MOTIONS. 

From  Oudemans'  paper  ("Astron.  Nach.,"  Aug.  1889),  with  some  modifications  from 
later  sources. 


No. 

NAME. 

a  (1900) 

6(1900) 

Mag. 

Parallax 

(P)- 

1 
i 

Distance 
(Light- 
Years) 
3.26. 
P 

Proper 
Motion  (fji). 

Cross 
Motion 
(miles 
p.  sec.) 

2.94^' 
P 

1 

a  Centauri  .... 

1#>32>°.6 

-60°  25' 

0.7 

0".75 

5 

4.4 

3".67 

14.4 

2 

LI.  21185      .... 

10  57.3 

+  36  56 

6.9 

0.50 

3 

6.5 

4.75 

28 

3 

61  Cygni     .... 

21     2.0 

+  38  12 

5.1 

0.40 

6 

8.1 

5.16 

37.9 

4 

T)  Herculis  .... 

17     9.6 

+  39     8 

3.7 

0.40? 

1 

8.1 

0.08 

0.6 

5 

Sirius     

6  40.4 

—  16  34 

-1.4 

0.39 

4 

8.4 

1.39 

10.5 

6 

S2398    ...... 

18  41.5 

+  59  28 

8.2 

0.35 

2 

9.3 

2.40 

20.3 

7 

f\  Cassiopeiae  .    .    . 

0  42.9 

+  57  18 

3.6 

0.35 

3 

9.3 

1.13 

9.5 

8 

/a  Cassiopeise  .    .    . 

1  01.0 

+  54  23 

5.2 

0.34 

1 

9.6 

3.75 

33.3 

9 

Groombridge  1618  . 

10     4.8 

+  50     1 

6.5 

0.32 

2 

10 

1.43 

13.2 

10 

v1  and  v2  Draconis  . 

17  30.3 

+  55  15 

4.9) 

4.8J 

0.30 

2 

10.9 

0.16 

1.6 

11 

Groombridge  34  .    . 

0  12.1 

+  42  24 

7.9 

0.29 

2 

11.2 

2.80 

28.4 

12 

Lac.  9352     .... 

22  58.8 

-36  29 

7.5 

0.28 

8 

11.6 

6.96 

73 

13 

Procyon     .... 

7  33.5 

+    5  30 

0.5 

0.27 

4 

12.1 

1.25 

13.6 

14 

LI.  21258     .... 

11     0.0 

+  44     5 

8.5 

0.26 

4 

12.5 

4.40 

49.8 

15 

Arg.-Oeltzen  11677  . 

11   14.4 

+  66  26 

9 

0.26 

2 

12.5 

3.04 

34.4 

16 

70Opbiuchi    .    .    . 

18     0.4 

+    2  33 

4.1 

0.25 

2 

13.2 

1.13 

13.3 

17 

<r  Draconis     .    .    . 

19  32.6 

+  69  28 

4.7 

0.25 

2 

13.2 

1.84 

21.7 

18 

e  Indi      . 

21  54.9 

—  57  14 

5.2 

0.20 

3 

16.3 

4.60 

67  7 

19 

a  Aquilse    .    . 

19  45.4 

+    8  35 

1 

0.20 

3 

16.3 

0.65 

Di  .* 

9.6 

20 

o2  Eridani  .... 

4  10.2 

-   7  49 

4.5 

0.19 

2 

17.2 

4.05 

62.7 

21 

Arg.-Oeltzen  17415-6 

17  34.1 

+  68  27 

9 

0.18 

2 

18.1 

1.27 

20.8 

22 

S  1516    . 

11     8.1 

+  74     4 

7 

0.17  ? 

I 

19.2 

0.42 

7  3 

23 

/3  Cassiopeia?  .    .    . 

0    3.3 

+  58  33 

2.4 

0.16 

2 

20.4 

0.55 

i  .0 
10 

24 

Vega 

18  33.5 

+  38  41 

0.2 

0.16 

3 

20.4 

0.36 

6.6 

25 

e  Eridani    .... 

3  15.5 

-43  29 

4.4 

0.14 

3 

23.3 

3.03 

63.7 

26 

Arcturus    .... 

14  10.6 

+  19  45 

0.3 

0.13 

1 

25.1 

2.28 

51.6 

27 

a  Tauri 

4  29.6 

+  16  17 

1 

0.116 

jj 

28.2 

0.19 

4.8 

28 

a  Aurigse    .... 

5     8.6 

+  45  53 

0.2 

0.107 

'2 

30.4 

0.43 

11.8 

29 

a  Leonis     .... 

10     2.5 

+  12  30 

1.4 

0.093 

2 

35.1 

0.27 

8.5 

30 

Groombridge  1830  . 

11  46.6 

+  38  31 

6.5 

0.087? 

1 

37.5 

7.05 

239    ? 

31 

Polaris 

1  18.5 

+  88  43 

2.1 

0.074 

g 

44 

0.045 

1.8 

32 

a  Cassiopeise  .    .    . 

0  34.3 

+  55  56 

2.2 

0.071 

1 

46 

0.05 

2.1 

33 

/3  Geminorum     .    . 

7  38.6 

+  28  17 

1.1 

0.068? 

1 

48 

0.64 

27.7 

34 

£Toucani  .... 

0  14.2 

-65  31 

4.1 

0.057 

2 

57 

2.05 

106 

35 

85Pegasi    .... 

23  56.4 

+  26  30 

5.8 

0.054 

1 

60 

1.29 

70.3 

Canopus,  a  Orionis,  a  Cygni,  0  Centauri,  and  y  Cassiopeise,  all  of  them  stars  of  the  first 
or  second  magnitude,  have  also  been  carefully  observed,  and  have  yielded  no  parallax 
exceeding  0".05. 

In  the  table  the  column  headed  "  weight  "  indicates  roughly  the  probable  reliability  of 
the  parallax  given,  — the  estimate  depending  on  the  character,  number,  and  accordance 
of  the  different  determinations  for  the  star  in  question.  The  average  "  probable  error  " 
for  the  parallaxes  of  the  table  may  be  taken  as  about  0".04,  i.e.  it  is  just  as  likely  as  not  that 
an  average  parallax,  weighted  2  or  3,  may  be  wrong  by  that  amount. 

The  original  paper  of  Oudemans  contains  all  the  data  then  available  :  in  the  cases  of 
several  of  the  stars  they  are  very  discordant  and  unsatisfactory,  so  that  it  is  to  be  expected 
that  ultimately  some  of  the  results  tabulated  above  will  prove  seriously  incorrect. 


608 


APPENDIX. 


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APPENDIX. 


609 


TABLE  VI.— VARIABLE   STARS. 

A  selection  from  Dr.  S.  C.  Chandler's  third  Catalogue  (July,  1896)  containing  such  as  are 
visible  to  the  naked  eye,  have  a  range  of  variation  exceeding  half  a  magnitude,  and  can  be 
seen  in  the  United  States. 


o' 

NAME. 

Place,  1900. 

Range  of 
Variation 

Period 

fc 

a 

S 

(Mag.). 

(Days). 

Xv6  m&rks  . 

1 

T.  Ceti    .    .    .    . 

Oh  16m.7 

-20°  37' 

5.1-  7.0 

65± 

Very  irreg. 

2 

R.  Andromedae    . 

0    18.8 

+  38     1 

5.6-12.8 

410.7 

3 

a  Cassiopeiae    .    . 

0    34.5 

+  55  59 

2.2-  2.8 

Not  periodic. 

4 

oCeti(Mira)   .    . 

2    14.3 

—  3  26 

1.7-  9.5 

331.6 

Large    irregu- 

larities in  date 

&  brightness. 

5 

P  Persei  .... 

2    58.8 

+  38  27 

3.4-  4.2 

33? 

Very  irreg. 

6 

/3  Persei  (Algol)  . 

3      1.7 

+  40  34 

2.3-  3.5 

2a20M8">55».43 

Period  now 

shortening. 

7 

ATauri   .... 

3    55.1 

+  12  12 

3.4-  4.2 

3d22h52»12» 

Algol  type. 

Irregular. 

8 

e  Aurigse      .    .    . 

4    54.8 

+  43  41 

3.0-  4.5 

Not  periodic. 

9 

a  Orionis     .    .    . 

5    49.7 

+   7  23 

0.7-  1.5 

Not  periodic. 

10 

rj  Geminorum 

6      8.8 

+  22  32 

3.2-  4.2 

231.4 

11 

£  Geminorum  .    . 

6    58.2 

+  20  43 

3.7-  4.5 

IQd  3h  41m  3Q8.6 

12 

R  Canis  Maj.   .    . 

7    14.9 

-16  12 

5.9-  6.7 

Id  3h  15m  468 

Algol  type. 

13 

R  Leonis  Min.     . 

9    39.6 

+  34  58 

6.0-13.0 

370.5 

14 

RLeonis     .    .    . 

9    42.2 

+  11  54 

5.2-10.0 

312.8 

15 

U  Hydras     .    .    . 

10    32.6 

-12  52 

4.5-  6.3 

195  ±? 

Very  irreg. 

16 

R  Ursas  Maj.   .    . 

10    37.6 

+  69  18 

6.0-13.2 

302.1 

17 

R  Hydras     .    .    . 

13    24.2 

-22  46 

3.5-  5.5 

425.15 

Period 

shortening. 

18 

S  Virginis    .    .    . 

13    27.8 

-   6  41 

5.7-12.5 

376.4 

19 

RBootis      .    .    . 

14    32.8 

+  27  10 

5.9-12.2 

223.4 

20 

S  Libras   .... 

14    55.6 

—  8     7 

5.0-  6.2 

2d  7h  51m  22'.8 

Algol  type. 

21 

R  Coronas    .    .     . 

15    44.4 

+  28  28 

5.8-13.0 

Not  periodic. 

22 

R  Serpentis     .    . 

15    46.1 

+  15  26 

5.6-13.0 

357.0 

23 

a  Herculis  .    .    . 

17    10.1 

+  14  30 

3.1-  3.9 

60  to  90* 

Not  periodic. 

24 

U  Ophiuchi     .    . 

17    11.5 

+    1   19 

6.0-  6.7 

20h  7m  428.56 

25 

u  Herculis  .    .    . 

17    13.6 

+  33  12 

4.6-  5.4 

Irreg.periodic. 

26 

X  Sagittarii     .    . 

17    41.3 

-27  48 

4.0-  6.0 

7d   OM7m57" 

27 

WSagittarii    .     . 

17    58.6 

-29  35 

4.8-  5.8 

7<J  14h  I6m  13s 

28 

Y  Sagittarii     .    . 

18    15.5 

-18  54 

5.8-  6.6 

5d  igh  33m  24«.5 

29 

RScuti   .... 

18    42.1 

-    5  49 

4.7-  9.0 

71.1 

Very  irreg. 

30 

/3  Lyras    .... 

18    46.4 

+  33  15 

3.4-  4.5 

12d  21h  47m  23".72 

31 

R  Lyras    .... 

18    52.3 

+  43  49 

4.0-  4.7 

46.4 

32 

xCygni  .... 

19    46.7 

+  32  40 

4.0-13.5 

406.02 

Period 

lengthening. 

33 

ij  Aquilas     .    .    . 

19    47.4 

+   0  45 

3.5-  4.7 

7d   41.  nm  593 

34 

S  Sagittas    .    .    . 

19    51.4 

+  16  22 

5.6-  6.4 

8d  9hllm48«.5 

35 

X  Cygni  .... 

20    39.5 

+  35  14 

6.4-  7.7 

I6d    9hl5m    78 

36 

T  Vulpeculas  .    . 

20    47.2 

+  27  53 

5.5-  6.5 

4d  iQh  27™  5Q8.4 

37 

TCephei     .    .    . 

21      8.2 

+  68     5 

5.2-10.7 

387 

38 

MCephei      .    .    . 

21    40.4 

+  58  19 

4.0-  5.5 

430  ± 

Irreg.periodic. 

39 

SCephei       .    .    . 

22    25.4 

+  57  54 

3.7-  4.9 

5d  8h  47m  39..3 

40 

/3  Pegasi  .... 

22    58.9 

+  27  32 

2.2-  2.7 

Not  periodic. 

41 

R  Aquarii    .    .    . 

23    38.6 

-15  50 

5.8-11? 

387.16 

42 

R  Cassiopeias  .    . 

23    53.3 

+  50  50 

4.8-12 

429.5 

610 


APPENDIX. 


8  § 
>  a 


5    a 


S 

" 


I    I    I    I    I    I    I    I    I    I    I    I  +  I  ++  I    I    I    I    I    I 


ceHcecececececeeerHrHcecerHcecece,OrHrHc3 

M  W  WWW  WWWMHHM  WMHrHWWMrHM  MM 


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11  .|1  .  .1 


+  4-  I  I  +  I  I  I  I  I 


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-»• 


APPENDIX. 


611 


TABLE  VIII.  —  MEAN  REFRACTION. 


Corresponding  to  temperature  of   50°  F.,  and  to  a  barometric  pressure  of 
29.6  inches. 


Altitude. 

Refraction. 

Altitude. 

Refraction. 

Altitude. 

Refraction. 

0° 

34'  50" 

11° 

4'47".7 

30° 

1'  39".  5 

1° 

24  22 

12° 

4  24  .5 

35° 

1   22  .1 

2° 

18  06 

13° 

4  04  .4 

40° 

1  08  .6 

3° 

14  13 

14° 

3  47  .0 

45° 

57  .6 

40 

11  37 

16° 

3  18  .2 

50° 

48  .3 

5° 

9  45 

18° 

2  55  .5 

55° 

40  .3 

6° 

8  23 

20° 

2  37  .0 

60° 

33  .2 

7° 

7  19 

22° 

2  21  .6 

65° 

26  .8 

8° 

6  29 

24° 

2  08  .6 

70° 

20  .9 

9° 

5  49 

26° 

1  57  .6 

80° 

10  .2 

10° 

5  16 

28° 

1  48  .0 

90° 

0  .0 

For  every  5°  F.  by  which  the  temperature  is  less  than  50°  F. ,  add  one  per 
cent  to  the  tabular  refraction,  and  decrease  it  in  the  same  ratio  for  temperatures 
above  50°  F. 

Increase  the  tabular  refraction  by  three  and  a  half  per  cent  for  every  inch  of 
barometric  pressure  above  29.6  inches,  and  decrease  it  in  the  same  ratio  below 
that  point.  These  corrections  for  temperature  and  pressure,  though  only  ap- 
proximate, will  give  a  result  correct  within  2"  except  in  extreme  cases. 


INDEX. 

See  also  the  Supplementary  Index. 

[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles  and  not  to  pages.] 


Aberration  of  light,  annual,99t22i=22fi4_ 
used  to  determine  the  solar  parallax, 
692;  diurnal,  226*;  spherical  and  chro- 
matic, 39. 

ABOUL  WEFA,  discoverer  of  the  lunar 
variation,  457. 

Absolute  scale  of  stellar  magnitude,  819. 

Acceleration  of  Ericke's  comet,  710;  of 
Winuecke's  comet,  711;  of  the  sun's 
equator,  283-285;  secular,  of  moon's 
mean  motion,  459-461;  secular,  of 
moon's  mean  motion  as  affected  by 
meteors,  778. 

Achromatic  object-glasses  for  telescopes, 
41. 

Actinometer  of  Violle,  341. 

ADAMS,  J.  C.,  the  discovery  of  Neptune, 
654;  investigation  of  the  orbit  of  the 
Leonids,  785. 

Adjustments  of  the  transit  instrument, 
60. 

Aerolites.    See  Meteorites. 

Age,  relative,  of  the  planets,  913,915;  of 
the  solar  system,  922;xof  the  sun,  359. 

Air-currents  at  high  elevations,  773,  note. 

AIRY,  G.  B.,  density  of  the  earth,  169. 

Albedo  denned  and  determined,  546;  of 
Jupiter,  614;  of  Mars,  583;  of  Mercury, 
558;  of  the  Moon,  259;  of  Neptune, 
660;  of  Saturn,  636;  of  Uranus,  648, 
of  Venus,  572. 

Algol,  or  £  Persei,  848. 

Almagest  of  Ptolemy,  500,  700, 795. 

Almucantar  defined,"  12. 

Altitude  defined,  21;  parallels  of,  12;  of 
pole  equals  latitude,  30;  of  sun,  how 
measured  with  sextant,  77. 

Altitude  and  azimuth  instrument,  71. 

Amplitude  defined,  22. 

Andromeda,  the  nebula  in,  886;  tempo- 
rary star  in  the  nebula  of,  845. 


Andromedes,  the,  780,  784,  786. 

Angle,  position,  of  a  double  star,  868;  of 

_ the  vertical,  156. 

Angular  and  linear  dimensions,  5;  veloc- 
ity under  central  force,  its  law,  408, 
409. 

Annual  equation  of  the  moon's  motion, 
458;  motion  of  the  sun,  172,  173. 

Annular  eclipse,  382;  nebula  in  Lyra, 
888. 

Anomalistic  month,  the,  397,  note ;  revo- 
lution of  the  moon,  250;  year  defined, 
216. 

Anomaly  defined,  mean  and  true,  189. 

Apertures,  limiting,  in  photometry,  825. 

Apex  of  the  sun's  way,  805. 

Apparition,  perpetual,  circle  of,  33. 

Apsides,  line  of,  defined,  183;  its  revolu- 
tion in  case  of  the  earth's  orbit,  199; 
its  revolution  in  case  of  the  moon's 
orbit,  454 ;  its  revolution  in  case  of  the 
planets'  orbits,  527. 

Arc  of  meridian,  how  measured,  147. 

Areal  or  areolar  velocity,  law  of,  under 
central  force,  402-406. 

Areas,  equable  description  of,  in  earth's 
orbit,  186,  187. 

ARGELANDER,  his  Durchmusterunf/  and 
zones,  795,  833 ;  his  star  magnitudes, 
817,  833. 

Argus,  r,,  841. 

Ariel,  the  inner  satellite  of  Uranus,  650. 

Aries,  first  of,  17. 

ARISTARCHUS,  method  of  determining  the 
sun's  distance,  666,  670. 

Artificial  horizon,  the,  78. 

Ashes  of  meteors,  775. 

Aspects  of  planets  defined  by  diagram, 
494. 

Asteroids,  the,  or  minor  planets,  592-601 ; 
theories  as  to  their  origin,  600. 

Astrsea,  the  fifth  asteroid,  discovered  by 
Hencke,  593. 


614 


INDEX. 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.] 


Astro -Physics  defined,  2. 

Atlases  of  the  stars,  793. 

Atmosphere  of  the  moon,  255-257;  of 
Venus,  573;  height  of  the  earth's,  98. 

Attraction,  intensity  of  the  solar,  on  the 
earth,  436;  within  a  hollow  sphere, 
169;  of  universal  gravitation,  161, 162. 

Axis  of  the  earth,  its  direction,  14 ;  of  the 
earth,  disturbed  by  precession,  206 ;  of 
the  sun,  its  direction,  282. 

Azimuth  defined,  22 ;  determination  of, 
127 ;  method  of  reckoning,  22 ;  of  tran- 
sit instrument,  its  adjustment,  60. 


BAILY,  determination  of  the  density  of 
the  earth,  166. 

Balance,  common,  used  in  determining 
the  density  of  the  earth,  170;  torsion, 
used  in  determining  the  density  of  the 
earth,  165. 

Barometer,  changes  of,  affecting  atmos- 
pheric refraction,  91 ;  effect  on  height 
of  the  tides,  480. 

Barometric  error  of  a  clock  and  its  com- 
pensation, 52. 

Beginning  of  the  day,  123;  of  the  year, 
222. 

BENZENBERG,  experiments  on  the  devia- 
tion of  falling  bodies,  138. 

BESSEL,  the  parallax  of  61  Cygni,  809, 811 ; 
formation  of  comets'  tails,  728;  his 
"zones,"  795. 

BIELA'S  comet,  744-746. 

Bielids,  the,  746,  780,  784,  786. 

Bielid  meteorite,  Mazapil,  784. 

Binary  stars,  872-879*;  their  masses, 
877-878;  their  orbits,  873-877:  spec- 
troscopic  binaries,  879-879*. 

Bissextile  year,  explanation  of  term,  219. 

Black  Drop,  the,  at  a  transit  of  Venus, 
681. 

BODE'S  law,  488,  489. 

Bolides,  or  detonating  meteors,  768. 

Bolometer,  the,  of  Langley,  343. 

BOND,  G.  P.,  first  photograph  of  a  double 
star,  868. 

BOND,  W.  C.,  discovery  of  Hyperion,  643; 
of  Saturn's  dusky  ring,  637. 

BOYLE,  law  of,  360,  note. 

BRAKE,  TYCHO.    See  TYCHO. 

BREDICHIN,  his  theory  of  comets'  tails, 
731,  732. 

Brightness  of  comets,  699 ;  of  planets  in 
various  positions,  Mercury,  551;  Ve- 
nus, 563,  568;  Mars,  579;  Asteroids, 
696,  599;  Jupiter,  610;  Saturn,  632; 


Uranus,  647 ;  Neptune,  660 ;  of  an  ob- 
ject in  the  telescope,  38;  of  shooting 
stars,  773 ;  of  stars,  causes  of  the  dif- 
ference in  this  respect,  836 ;  of  stars, 
its  measurement,  823-831. 

C. 

Calendar,  the,  217-223. 

Callisto,  the  outer  satellite  of  Jupiter, 

621,  627. 

Calories  of  different  magnitude,  338,  note. 
Candle  power,  its  mechanical  equivalent, 

776 ;  power  of  sunlight,  332,  333. 
Candle  standard,  333,  note. 
Capture  theory  of  comets,  740. 
Cardinal  points  defined,  20. 
CARLINI,  earth's  density,  168. 
CARRINGTON,  law  of  the  sun's  rotation, 

283,  284. 

CASSEGRAINIAN  telescope,  48. 
CASSINI,  J.  D.,  discovery  of  the  division 

in  Saturn's  ring,  637;    discovery  of 

four  satellites  of  Saturn,  643. 
Catalogues  of  stars,  795. 
CAVENDISH,  the  torsion  balance,  165. 
Celestial  latitude  and  longitude,  178, 179; 

sphere,  conceptions  of  it,  4. 
Cenis,  Mt.,  determination  of  the  earth's 

density,  168. 
Central  force,  motion  under  it,  400-410; 

force,  its  measure  in  case  of  circular 

motion,  411. 
Central  suns,  807,  903. 
Centrifugal  force  of  the  earth's  rotation, 

154. 

Ceres,  discovery  of,  592. 
CHANDLER,  S.  C.,  catalogue  of  variable 

stars,  852,  Appendix,  Table  VI. 
Changes  on  the  moon's  surface,  268;  in 

the  nebulae,  892. 
Characteristics    of    different   meteoric 

swarms,  783. 
Charts  of  the  stars,  798. 
Chemical  elements  recognized  in  comets, 

724,725;  elements  recognized  in  stars, 

856;  elements  recognized  in  the  sun, 

315-317. 

Chromatic  aberration  of  a  lens,  39. 
Chromosphere,  the,  291,  322,  363. 
Chronograph,  the,  56. 
Chronometer,  the,  54;  longitude  by,  121 

[A]. 

Circle,  the  meridian,  63. 
Circles  of  perpetual  apparition  and  oc- 

cultation,  33. 

Circular  motion,  central  force  in,  411. 
CLAIRAUT'S    equation    concerning    the 

ellipticity  of  the  earth,  155. 


INDEX. 


615 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.} 


CLARKE,  COL.,  dimensions  of  the  earth, 
145  and  Appendix. 

Classification  of  stellar  spectra,  857, 858. 

CLERKE,  Miss  A.  M.,  her  history  of  as- 
tronomy, Preface, 
746,900. 

Clocks,  general  remarks  on,  50. 

Clock-breaks  (electric),  57. 

Clock-error,  or  correction,  and  rate,  53; 
or  correction  determined  by  transit 
instrument,  59. 

Clusters  of  stars,  883-886. 

COGGIA'S  Comet,  730. 

Coliimating  eye-piece,  67. 

Collimation  of  transit  instrument,  60. 

Collimator,  the,  used  with  transit  instru- 
ment, 60;  of  a  spectroscope,  311. 

Collision  theory  of  variable  stars,  850. 

Colors  of  stars  in  photometry,  830;  of 
double  stars,  867. 

Colures  defined,  25. 

Comet,  Biela's,  744;  Donati's,  727,  730, 
747;  Encke's,  710,  743;  great,  of  1882, 
748-752;  Halley's,  742;  Winnecke's, 
711. 

Comets,  acceleration  of  Encke's  and  Win- 
necke's, 710,  711;  brightness  of,  699, 
723;  capture  theory  of,  740;  chemical 
elements  in,  724, 725 ;  constituent  parts, 
713;  contraction  of  head  when  near  the 
sun,  715 ;  danger  from,  753,  754 ;  den- 
sity of,  720;  designation  of,  697;  di- 
mensions of,  714, 717 ;  ejection  theory, 
741 ;  fall  upon  earth  or  sun,  probable 
effect,  754;  groups  of,  with  similar 
orbits,  705;  their  light,  721;  their 
masses,  718,  719;  and  meteors,  their 
connection,  785-787;  nature  of,  737; 
their  orbits,  700-709;  origin  of,  738- 
741;  perihelia,  distribution  of,  706; 
physical  characteristics,  712;  plane- 
tary families  of,  739;  their  spectra, 
724, 726 ;  superstitions  regarding  them, 
695;  their  tails  or  trains,  713,  717,  728- 
736 ;  variations  in  brightness,  723 ;  vis- 
itors in  the  solar  system,  709. 

Comparison  of  starlight  with  sunlight, 
334,  832. 

Compensation  pendulums,  51. 

Compensation  of  pendulum  for  barome- 
tric changes,  52. 

Components  of  the  disturbing  force,  445. 

Co-ordinates,  astronomical,  20. 

COMMON,  A.  A.,  photographs  of  nebulae, 
893. 

Conies,  the,  422,  423. 

Connection  between  comets  and  meteors, 
785-787. 


Constant  of  aberration,  the,  225  ;  the  so- 

lar, 338-340. 
Constancy,  secular,  of  the  mean  distances 

and  periods  of  the  planets,  526. 
Constellations,  list  of,  792  ;  their  origin, 

791. 
Contact  observations,  transit  of  Venus, 

679-682. 

Contraction  theory  of  solar  heat,  356. 
Conversion  of  R.  A.  and  Decl.  to  latitude 

and  longitude,  180. 
COPERNICUS,  his  system,  503;   "Trium- 

phans,"  809. 
CORNU,  determination  of  the  earth's  den- 

sity, 166  ;  photometric  observation  of 

eclipses  of  Jupiter's  satellites,  630. 
Corona,  the  solar,  291,  327-331,  364. 
Cosmogony,  905-917. 
Cotidal  lines,  475. 
Craters  on  the  moon,  265-267. 
CREW,  H.,  spectroscopic  observations  of 

the  sun's  rotation,  285,  note. 
Crust  of  meteorites,  761. 
Curvature  of  comet's  tails,  729. 
Curvilinear  motion  the  effect  of  force,  401. 
Cycle,  the  Metonic,  218  ;  the  Callipic,  218. 
Cyclones  as  proofs  of  the  earth's  rotation, 

143. 


DALTON,  his  law  of  gaseous  mixtures,  360, 

note. 

Danger  from  comets,  753,  754. 
Darkening  of  the  sun's  limb,  337. 
DARWIN,  G.  H.,  rigidity  of  the  earth,  171  ; 

tidal  evolution,  484,  916. 
DAWES,  diameter  of  the  spurious  discs 

of  stars,  43  ;  nucleoli  in  sun  spots,  293. 
Day,  the  civil  and  the  astronomical,  117  ; 

effect  of  tidal  friction  upon  its  length, 

461  ;  changes  in  its  length,  144  ;  where 

it  begins,  123. 
Declination  denned,  23;  parallels  of,  23; 

determined  with  the  meridian  circle, 

128. 
Degree  of  the  meridian,  how  measured, 

135,  147. 
Deimos,  the  outer  satellite  of  Mars,  590, 

591. 
DELISLE,  method  of  determining  the  so- 

lar parallax,  682. 
DENNISG,  drawings  of  Jupiter's  red  spot, 

618. 
Density  of  comets,  720  ;  of  the  earth,  de- 

terminations of    it,  164-170;    of    the 

moon,  246;  of  a  planet,  how  deter- 

mined, 540;  of  the  sun,  279. 
Detonating  meteors,  or  "  Bolides,"  768. 


616 


INDEX. 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.} 


Development  of  sun  spots,  297. 

Dhurmsala  meteorite,  ice-coated,  765. 

Diameter  (apparent)  as  related  to  dis- 
tance, 6 ;  of  a  planet,  how  measured, 
534. 

Differential  method  of  determining  a 
body's  place,  129;  method  of  deter- 
mining stellar  parallax,  811. 

Diffraction  of  an  object-glass,  43. 

Dione,  fourth  satellite  of  Saturn,  643,  note. 

Dip  of  the  horizon,  81. 

Disc,  spurious,  of  stars  in  telescope,  43. 

Discovery  of  comets,  698. 

Dissipation  of  energy,  925. 

Distance  of  the  moon,  239;  of  the  nebu- 
Ise,  896;  and  parallax,  relation  be- 
tween, 84;  of  a  planet  in  astronomi- 
cal units,  how  determined,  515-518 ;  of 
the  stars,  808-815;  of  the  sun,  274, 
275,  also  Chap.  XVI. 

Distinctness  of  telescopic  image,  its  con- 
ditions, 39. 

Distribution  of  the  nebulae,  895;  of  the 
stars,  899;  of  the  sun  spots,  301. 

Disturbing  force,  the,  439-444;  force, 
diagram  of,  441 ;  force,  its  resolution 
into  components,  445. 

Diurnal  aberration,  226*;  inequality  of 
the  tides,  471;  parallax,  82,86;  phe- 
nomena in  various  latitudes,  191. 

Divisions  of  astronomy,  2. 

DOEBFEL  proves  that  a  comet  moves  hi  a 
parabola,  700. 

DONATI'S  comet,  727,  730,  747. 

DOPPLEB'S  principle,  321*. 

Double  stars,  866-S79;  their  colors,  867; 
criterion  for  distinguishing  between 
those  optically  and  physically  double, 
870;  method  of  measuring  them,  868; 
optically  and  physically  double,  869; 
having  orbital  motion,  see  Binary 
Stars. 

DRAPER,  H.,  oxygen  in  the  sun,  316 ;  pho- 
tograph of  the  nebula  in  Orion,  893; 
photography  of  stellar  spectra,  859; 
memorial,  the,  859. 

Duration,  future,  of  the  sun,  358 ;  of  sun 
spots,  300. 


Earth,  the,  her  annual  motion  proved  by 
aberration  and  stellar  parallax,  174 ; 
approximate  dimensions,  how  meas- 
ured, 134-5-6;  constitution  of  its  in- 
terior, 171;  its  dimensions,  Appendix 
and  145 ;  its  dimensions  determined  ge- 
odetically,  147-149 ;  form  of,  from  pen- 
dulum experiments,  152-155;  growth 


of,  by  accession  of  meteoric  matter} 
777 ;  mass  compared  with  that  of  the 
sun,  278 ;  its  mass  and  density,  159- 
170;  its  orbit,  form  of,  determined, 
182;  principal  facts  relating  to  it,  132 ; 
proofs  of  its  rotation,  138-143. 

Earth-shine  on  the  moon,  254. 

Eccentricity  of  the  earth's  orbit,  discov- 
ered by  Hipparchus,  184 ;  of  the  earth's 
orbit,  how  determined,  185;  of  the 
earth's  orbit,  secular  change  of,  198 ; 
of  an  ellipse  denned,  183,  506. 

Eclipses,  duration  of  lunar,  373;  dura- 
tion of  solar,  385 ;  number  in  a  year, 
391-393 ;  recurrence  of,  the  saros,  395 ; 
of  Jupiter's  satellites,  627-630 ;  of  the 
moon,  370-378;  of  the  sun,  379-390; 
total,  of  the  sun,  as  showing  the  solar 
atmosphere  and  corona,  319,  323. 

Ecliptic,  the,  denned,  175 ;  obliquity  of, 
176;  limits,  lunar,  374,  375;  limits,  so- 
lar, 386. 

Effective  temperature  of  the  sun,  351. 

Ejection  theory  of  comets  and  meteors, 
741. 

Elbowed  equatorial,  the,  74. 

Electrical  registration  of  observations,56. 

Electro-dynamic  theory  of  gravitation, 
602. 

Elements,  chemical,  not  truly  elemen- 
tary, 318;  chemical,  recognized  in 
comets,  724, 725 ;  chemical,  recognized 
in  stars,  856 ;  chemical,  recognized  in 
sun,  316, 317;  of  a  planet's  orbit,  505- 
508. 

ELKIN,  stellar  parallaxes,  808,  814,  815, 
and  Appendix,  Table  IV. 

Ellipse  denned,  183 ;  described  as  a  conic, 
422,  423. 

Elliptic  comets,  their  number,  702 ;  their 
orbits,  703 ;  recognition  of,  704. 

Ellipticity  or  oblateness  of  a  planet  de- 
fined, 150. 

Elongation  of  moon  or  planet  defined,  230. 

Enceladus,  the  second  satellite  of  Saturn, 
643,  note. 

ENCKE'S  comet,  710,  743. 

ENCKE,  his  reduction  of  the  transits  of 
Venus,  667. 

Energy,  the  dissipation  of,  925 ;  and  work 
of  solar  radiation,  345. 

Enlargement,  apparent,  of  bodies  near 
horizon,  4,  note,  88,  93. 

Envelopes  in  the  head  of  a  comet,  713, 727. 

Epoch  of  a  planet's  orbit  defined,  508. 

Epsilon  Lyrse,  653,  866,  882. 

Equal  altitudes,  determination  of  time, 
115. 


INDEX. 


617 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.] 


Equation,  annual,  of  moon's  motion,  458; 
of  the  centre,  189;  of  the  equinoxes, 
213;  of  light,  by  means  of  Jupiter's 
satellites,  628-630;  of  time  explained, 
201-204;  expressing  the  relation  be- 
tween the  light  of  different  stellar 
magnitudes,  820. 

Equator,  the  celestial,  16. 

Equatorial  acceleration  of  the  sun's  ro- 
tation, 28:5-285 ;  conde',  Paris,  74 ;  par- 
allax, 85 ;  telescope,  72 ;  telescope  used 
to  determine  the  place  of  a  heavenly 
body,  121). 

Equinoctial,  the,  see  Equator,  celestial ; 
points,  or  equinoxes,  17. 

Equinoxes,  the,  equation  of,  213;  preces- 
sion of,  205-212. 

ERATOSTHENES,  his  measure  of  the  earth, 
135. 

Erecting  eye-piece  for  telescope,  45. 

ERICSSON,  his  solar  engine,  345;  experi- 
ment upon  radiation  of  molten  iron, 
350. 

Eruptive  prominences,  325. 

Escapement  of  clock,  50. 

Establishment  of  a  port  (harbor)  de- 
fined, 463. 

Europa,  the  second  satellite  of  Jupiter,621. 

Evection,  the,  456. 

Evolution,  tidal,  484,  916. 

Eye-pieces,  telescopic,  44. 

Extinctions,  the  method  of,  in  photome- 
try, 825. 

K. 

Faculse,  solar,  292. 

Fall  of  a  planet  to  the  sun,  time  required, 
413,  3 ;  of  a  comet  on  the  sun,  proba- 
ble effect,  754. 

Falling  bodies,  eastward  deviation,  138. 

Families  (planetary)  of  comets,  739. 

FAYE,  H.  A.,  his  modification  of  the  neb- 
ular hypothesis,  915;  theory  of  sun 
spots,  304. 

Flattening,  apparent,  of  the  celestial 
sphere,  4,  note. 

Force,  evidenced  not  by  motion,  but  by 
change  of  motion,  400;  projectile, 
term  carelessly  used,  401 ;  central,  mo- 
tion under  it,  400-410 ;  repulsive,  action 
on  comets,  728-733. 

Form  of  the  earth,  145-155. 

Formation  of  comets'  tails,  728. 

FOUCAULT,  the  gyroscope,  showing  earth's 
rotation,  142;  his  pendulum  experi- 
ment, showing  earth's  rotation,  139- 
141 ;  measures  velocity  of  light,  690. 

Fourteen  hundred  and  seventy-four  line 
of  the  spectrum  of  the  corona,  329. 


FRAUNHOFER  lines  in  the  solar  spectrum, 
315, 855 ;  observations  on  stellar  spec- 
tra, 855. 

Free  wave,  velocity  of,  473. 

Frequency,  relative,  of  solar  and  lunar 
eclipses,  394. 

O. 

Galaxy,  the,  898. 

GALILEO,  discovery  of  Jupiter's  sat- 
ellites, 621;  discovery  of  Saturn's 
rings,  637;  discovery  of  phases  of 
Venus,  567 ;  use  of  pendulum  in  time- 
keeping, 50. 

GALLE,  optical  discovery  of  Neptune, 
654. 

Ganymede,  the  third  satellite  of  Jupiter, 
621. 

Gas  contracting  by  loss  of  heat,  Lane's 
law,  357. 

GAUSS,  computes  the  orbit  of  Ceres,  592 ; 
determination  of  the  elements  of  an 
orbit,  519;  peculiar  form  of  achroma- 
tic object-glass,  41. 

GAY  LUSSAC,  law  of  gaseous  expansion, 
360,  note. 

Geocentric  latitude,  156;  place  of  a 
heavenly  body,  511. 

Geodetic  determination  of  the  earth's 
dimensions,  147,  149. 

Genesis  of  the  solar  system,  908-915 ;  of 
star  clusters  and  nebulae,  924. 

Georgium  Sidus,  the  original  name  for 
Uranus,  645. 

GILL,  solar  parallax  from  observations 
of  Mars,  676 ;  stellar  parallaxes,  808. 
Appendix,  Table  IV. 

Globe,  celestial,  rectification  of,  33,  note. 

Gnomon,  determination  of  latitude  with 
it,  107 ;  determination  of  the  obliquity 
of  the  ecliptic,  176. 

Golden  number,  the,  218. 

Gradual  changes  in  the  light  of  the  stars, 
839. 

Graduation  errors  of  a  circle,  69. 

Grating  diffraction,  311,  note. 

Gravitation,  electro-dynamic,  theory  of, 
602;  law  stated,  161;  nature  unknown, 
161;  law  extending  to  the  stars,  872, 
note,  873,  901,  note ;  Newton's  verifi- 
cation of  the  law  by  means  of  the 
moon's  motion,  419,  420. 

Gravitational  astronomy  defined,  2; 
methods  of  determining  the  solar  par- 
allax, 687-689. 

Gravity,  increase  of,  below  the  earth's 
surface,  169;  variation  of,  between 
equator  and  pole,  152.* 


618 


INDEX. 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.'] 


GREGORIAN  calendar,  the,  and  its  adop- 
tion in  England,  220,  221 ;  telescope,  48. 

Groups,  cometary,  705;  of  stars  having 
common  motion,  803. 

Growth  of  the  earth  by  meteoric  matter, 
777. 

Gyroscope,  Foucault's  proof  of  earth's 
rotation,  142 ;  illustrating  the  preces- 
sion of  the  equinoxes,  210,  211. 


HALL,  A.,  discovery  of  the  satellites  of 
Mars,  590 ;  on  the  question  whether  it 
is  certain  that  gravitation  extends 
through  the  stellar  universe,  901,  note. 

HALLEY,  his  comet,  742 ;  his  computation 
of  cometary  orbits,  700;  his  method 
of  determining  the  sun's  parallax,  679, 
680;  the  moon's  secular  acceleration, 
459 ;  proper  motions  of  stars,  800. 

HANSEN,  correction  of  the  solar  parallax, 
667 ;  opinion  on  the  form  of  the  moon, 
258. 

HARDING  discovers  Juno,  593. 

HARKNESS,  observations  on  the  light  of 
meteors,  776 ;  observation  of  the  cor- 
ona spectrum,  329. 

Harmonic  law,  Kepler's,  412-417. 

Harton  colliery,  density  of  the  earth,  169. 

Harvard  photometry,  the,  827,  828. 

Harvest  and  hunter's  moons  explained, 
237. 

Heat  and  light  of  meteors  explained,  765  ; 
of  the  moon,  260 ;  of  the  sun,  338-358 ; 
received  by  the  earth  from  meteors, 
355,  779;  from  the  stars,  834. 

Height  of  lunar  mountains,  270. 

HEIS,  enumeration  of  naked-eye  stars,  818. 

Heliocentric  place  of  a  planet,  512. 

Heliometer,  the,  677  j  used  in  determin- 
ing solar  parallax,  676,  683 ;  used  in 
determining  stellar  parallax,  811,  815. 

Helioscopes,  or  solar  eye-pieces,  286, 287. 

Helium,  a  recently  identified  gas  in  the 
solar  chromosphere,  323. 

HELMHOLTZ,  contraction  theory  of  solar 
heat,  356. 

HENCKE,  discovers  Astraea,  the  fifth  aste- 
roid, 593. 

HENDERSON,  measures  the  parallax  of 
a  Centauri,  809,  810. 

HENRY  BROTHERS,  astronomical  pho- 
tography, 272,  798. 

HENRY,  PROF.  J.,  heat  of  sun  spots,  310 ; 
at  sun's  limb,  348. 

HERSCHEL,  SIR  JOHN,  astrometry,  819; 
illustration  of  the  planetary  system, 


HERSCHEL,  SIR  W.,  discovery  of  the  sun's 
motion  in  space,  80-1- ;  discovery  of  two 
satellites  of  Saturn,  643 ;  discovery  of 
Uranus,  645;  discovery  of  two  satel- 
lites of  Uranus,  650;  star-gauges,  899; 
theory  of  sun  spots,  302;  his  reflect- 
ing telescope,  48. 

HEVELIUS,  his  view  of  cometary  orbits, 
700. 

HIPPARCHUS,  discovers  eccentricity  of 
earth's  orbit,  184 ;  discovers  precession, 
205;  his  value  of  the  solar  parallax, 
671 ;  the  first  star-catalogue,  795. 

HOLDEN,  E.  S.,  on  changes  in  nebulae, 
892. 

Horizon,  apparent  enlargement  of  bodies 
near  it,  4,  note,  88,  93;  artificial,  78; 
rational  and  apparent  defined,  10; 
dip  of,  81 ;  visible,  defined,  11. 

Horizontal  parallax,  83,  84 ;  point  of  the 
meridian  circle,  67. 

Hour-angle  defined,  24. 

Hour-circle  defined,  18. 

HUGGINS,  SIR  W.,  attempts  to  photograph 
the  corona  without  an  eclipse,  328; 
attempted  observation  of  stellar  heat, 
834;  observations  of  stellar  spectra, 
856;  photography  of  stellar  spectra, 
859;  spectroscopic  observations  of  T 
coronse,  844;  star-motions  in  line  of 
sight,  802 ;  spectrum  of  nebulae,  890. 

HUMBOLDT,  A.  VON,  classification  of  the 
planets,  549. 

HUNT,  STERRY,  carbonic  acid  brought  to 
earth  by  comets,  735. 

HUYGHENS,  discovery  of  Saturn's  rings, 
637;  discovery  of  Saturn's  satellite, 
Titan,  643 ;  invention  of  the  pendulum 
clock,  50;  his  long  telescope,  40. 

Hydrogen  in  the  solar  chromosphere  and 
prominences,  323-325 ;  bright  lines  oi 
its  spectrum  in  the  nebulae,  890 ;  bright 
lines  of  its  spectrum  in  temporary 
stars,  844 ;  bright  lines  of  its  spectrum 
in  variable  stars,  857. 

Hyperbola,  the,  described  as  a  conic, 
422. 

Hyperbolic  comets,  702. 

Hyperion,  the  seventh  and  last  discovered 
satellite  of  Saturn,  643,  644. 

Hypothesis,  nebular.  See  Nebular  hy- 
pothesis. 

I. 

lapetus,  the  outermost  satellite  of  Sat- 
urn, 643. 

IBN  JOUNIS,  use  of  pendulum  in  observa- 
tion, 50. 


INDEX. 


619 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.] 


Ice,  amount  melted  by  solar  radiation, 

344-346,  364*. 
Illumination  of  the  moon's  disc  during 

a  lunar  eclipse,  376. 
Image,  telescopic,  conditions  of  distinct- 

ness, 39. 

Inequality,  diurnal,  of  the  tides,  471. 
Inferences  deducible  from  Kepler's  laws, 

418. 

Inferior  planet,  motion  of,  498-9. 
Infinity,  velocity  from,  272*,  429,  435, 

1008-9. 

Influences  of  the  moon  on  the  earth,  262. 
Intra-Mercurial  planets,  602-606  ;  plan- 

ets, supposed  observations  of,  during 

solar  eclipse,  605. 

Interior  of  the  earth,  its  constitution,  171. 
Invariable  plane  of  the  solar  system,  531. 
Invariability  of  the  earth's  rotation,  144. 
lo,  the  first  satellite  of  Jupiter,  621. 
Iron  meteorites,  758  ;  in  the  sun,  315. 
Irradiation    in  micrometric   measures, 

256,  534. 

J. 

JANSSEN,  discovery  of  the  spectroscopic 
method  of  observing  the  solar  promi- 
nences, 323  ;  solar  photography,  289. 

JOLLY,  observations  of  the  earth's  den- 
sity, 170. 

Julian  calendar,  219. 

Juno,  discovered  by  Harding,  593. 

Jupiter,  the  planet,  609-631  ;  brightness 
as  seen  from  a  Centauri,  881  ;  a  semi- 
sun,  619  ;  his  comet-family,  739. 

K. 

KANT,  proposes  the  nebular  hypothesis, 
908. 

KELVIN,  LORD,  see  Thomson,  Sir  W. 

KEPLER,  his  belief  as  to  cometary  orbits, 
700  ;  his  three  laws  of  planetary  mo- 
tion, 412-418  ;  his  "  problem,"  188  ;  his 
"regular  solid  "  theory  of  the  planet- 
ary distances,  592,  note. 

KIRCHHOFF,  his  fundamental  principles 
of  spectrum  analysis,  314. 


LANGLEY,  S.  P.,  his  Bolometer,  343;  the 
color  of  the  sun,  337  ;  observations  on 
lunar  heat,  260,  261  ;  on  solar  heat,  348  ; 
sun-spot  drawings,  292;  light  of  sun 
spots,  293  ;  heat  of  sun  spots,  301*. 

LANE'S  law,  rise  of  temperature  conse- 
quent on  the  contraction  of  a  gaseous 
mass,  357. 


LA  PLACE,  his  equations  relating  to  the 
eccentricities  and  inclinations  of  the 
planetary  orbits,  532;  explanation  of 
the  moon's  secular  acceleration,  459, 
460;  the  invariable  plane  of  the  solar 
system,  531;  the  nebular  hypothesis, 
901-911. 

LASSELL,  discovery  of  the  two  inner  sat- 
ellites of  Uranus,  650;  independent 
discovery  of  Hyperion,  643. 

Latitude  (astronomical)  of  a  place  on  the 
earth's  surface,  30, 100, 156 ;  astronom- 
ical, geodetic,  and  geocentric,  dis- 
tinguished, 156;  determination  of, 
methods  used,  101-107 ;  at  sea,  its  de- 
termination, 103;  possible  variations 
of  it,  108;  station  errors,  158;  celes- 
tial, defined,  178,  179;  and  longitude 
(celestial),  conversion  into  o  and  5, 
180 ;  motion  of  planets  in,  495. 

Law  of  angular  velocity  under  central 
force,  408;  Bode's,  488,489;  of  Boyle 
or  Mariotte,  density  of  a  gas,  360,  note ; 
of  Dalton,  mixture  of  gases,  360,  note ; 
of  earth's  orbital  motion,  186,  187;  of 
equal  areas,  186,  402-406,  412 ;  of  Gay 
Lussac,  gaseous  expansion,  360,  note ; 
of  gravitation,  161,  162,  419,  872; 
Lane's,  of  temperature  in  gaseous  con- 
traction, 357;  of  linear  velocity  in 
angular  motion,  407. 

Laws  of  Kepler,  412^418;  motion  under  a 
central  force,  400-411. 

Leap  year,  rule  for,  220. 

Length,  of  the  day,  possible  changes  in  it, 
144 ;  of  the  year,  its  invariability,  526, 
778. 

Leonids,  the,  780,  786. 

LESCARBAULT,  supposed  discovery  of 
Vulcan,  603. 

Level  adjustment  of  the  transit  instru- 
ment, 60. 

LEVERRIER,  discovery  of  Neptune,  653, 
654 ;  on  an  intra-Mercurial  planet,  603 ; 
motion  of  the  perihelion  of  Mercury's 
orbit,  602 ;  method  of  determining  the 
solar  parallax  by  planetary  perturba- 
tions, 689. 

LEXELL'S  comet,  approach  to  Jupiter,, 
718;  recognition  of  Uranus  as  a 
planet,  645. 

Librations  of  the  moon,  249,  250,  251. 

Light  of  comets,  722;  of  the  moon,  259; 
emitted  by  certain  stars,  835 ;  received 
by  the  earth  from  certain  stars,  832; 
of  the  sun,  332-337 ;  total,  of  the  stars, 
833;  equation  of,  from  Jupiter's  satel- 
lites, 628-630;  mechanical  equivalent 


620 


INDEX. 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.} 


of,  Thomsen,  776;  time  occupied  by, 
in  coming  from  the  sun,  275,  629;  ve- 
locity of,  225,  note,  668,  690. 

Light-curves  of  variable  stars,  84S. 

Light-gathering  power  of  telescopes,  38. 

Light-ratio,  the,  in  scale  of  star-magni- 
tudes, 819. 

Light-year,  the,  the  unit  of  stellar  dis- 
tance, 814. 

Limb  of  the  sun,  darkening  of,  337;  of 
the  sun,  diminution  of  heat,  348. 

Limiting  apertures  in  stellar  photometry, 
825. 

Linear  and  angular  dimensions,  their  re- 
lation, 5 ;  velocity  under  central  force, 
its  law,  407,  409. 

Linne,  lunar  crater  supposed  to  have 
changed,  269. 

LISTING,  dimensions  of  the  earth,  145. 

Local  and  standard  time,  122. 

LOCKYBR,  SIR  J.  N.,  discovery  of  spec- 
troscopic  method  of  observing  the 
solar  prominences,  323;  his  "collision 
theory  "  of  variable  stars,  850;  views 
as  to  the  compound  nature  of  the  so- 
called  chemical  "elements,"  318; 
origin  of  the  Fraunhofer  lines,  320; 
theory  of  sun  spots,  306;  meteoric 
theory  of  nebulae,  894;  meteoric  hy- 
pothesis, 926. 

LOEWY,  peculiar  method  of  determining 
the  refraction,  95. 

Longitude,  arcs  of,  to  determine  the 
earth's  dimensions,  151;  (terrestrial), 
determination  of,  118-121 :  (celestial) , 
178-180;  of  perihelion,  505,  506. 

Luminosity  of  bodies  at  low  tempera- 
tures, 737,  note. 

Lunar  distances,  120,  B;  eclipses,  370- 
378;  influences  on  the  earth,  262; 
methods  of  determining  the  longitude, 
120 ;  perturbations,  448-461 ;  perturba- 
tions used  to  determine  the  solar  par- 
allax, 687. 

Lyrae,  a,  see  Vega;  /3,  variable  star,  847; 
e,  quadruple  star,  653,  866,  882. 

M. 

MADLER,   speculations  as  to  a  central 

sun,  807,  903. 
Magnifying  power  of  a  telescope,  37; 

power,  highest  available,  43. 
Magnitudes  of  stars,  816-822. 
Magnitude  of  smallest  star  visible  in  a 

given  telescope,  822. 
Magnesium  in  the  nebulae,  890,  894. 
Maintenance  of  the  solar  heat,  353-356. 


Mars,  the  planet,  578-591 ;  observed  for 
solar  parallax,  673-677. 

MASKELYNE,  his  mountain  method  of 
determining  the  earth's  density,  164. 

Mass  and  weight,  distinction  between 
them,  159,  160;  of  comets,  718,  719; 
of  the  earth  compared  with  the  sun, 
278;  of  the  earth  in  terms  of  the 
sun  as  determining  the  solar  parallax, 
689;  of  the  moon,  its  determination, 
243 ;  of  a  planet,  how  determined,  536- 
539;  of  the  sun,  compared  with  the 
earth,  278 ;  probable,  of  shooting  stars, 
776. 

Masses  of  binary  stars,  877,  878. 

MAYER,  R.,  meteoric  theory  of  the  solar 
heat,  353. 

MAXWELL,  CLERK,  meteoric  theory  of 
Saturn's  rings,  641. 

Mazapil,  the  meteorite  of,  784. 

Mechanical  equivalent  of  light,  776. 

MENDENHALL,  T.  C.,  pendulum  for  grav- 
ity determinations,  152*. 

Mercury,  the  planet,  551-562;  motion  of 
its  perihelion,  602. 

Meridian,  the  celestial,  defined,  19;  arc 
of,  how  measured,  147. 

Meridian-circle,  the,  63;  used  to  deter- 
mine the  place  of  a  heavenly  body, 
128. 

Meridian  photometer,  the,  828. 

Meteors  and  shooting  stars,  755-787 ;  ashes 
of,  775;  and  comets,  their  connection, 
785-787 ;  daily  number  of,  771 ;  detonat- 
ing, 768;  effect  on  the  earth's  orbital 
motion,  778;  effect  upon  the  moon's 
motion,  778;  effect  upon  the  trans- 
parency of  space,  779;  explanation 
of  their  light  and  heat,  765;  heat 
from  them,  355,  779;  magnitude  of, 
762;  method  of  observing  them,  764; 
their  trains,  766. 

Meteoric  growrth  of  the  earth,  777 ;  show- 
ers, 780-786;  shower  of  the  Bielids, 
1872, 1886, 746 ;  shower  of  the  Leonids, 
1833,  1866-67,  781 ;  swarms  and  rings, 
783;  swarms,  special  characteristics, 
783;  theory  of  Saturn's  rings,  641; 
theory  of  the  sun's  heat,  353-355; 
theory  of  sun  spots,  306. 

Meteorites,  or  uranoliths,  or  aerolites, 
755-769;  chemical  elements  in  them, 
760 ;  crust,  761 ;  fall  of,  756 ;  which 
have  fallen  in  the  United  States,  759 ; 
iron,  list  of,  758 ;  number  of,  769 ;  ques- 
tion of  their  origin,  767 ;  their  paths, 
763. 

Metonic  cycle,  the,  218. 


INDEX. 


621 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.} 


MICHELSON,  determination  of  the  veloc- 
ity of  light,  225,  note,  668,  690. 

Micrometer,  the  filar,  73,  534,  867. 

Microscope,  the  reading,  64. 

Midnight  sun,  the,  191. 

Milky  Way,  or  galaxy,  898. 

Mimas,  the  innermost  satellite  of  Saturn, 
643  and  note.  j 

Minor  planets,  or  asteroids,  592-601. 

Mira,  Omicron  Ceti,  846. 

Missing  stars,  840. 

Mohammedan  calendar,  217. 

Monocentric  eye-piece  for  telescope,  45. 

Moon,  the,  227-272;  her  atmosphere,  255- 
258;  the,  regarded  as  a  clock,  120; 
distance  of,  etc.,  240;  her  heat  and 
temperature,  260,  261;  her  light  as 
compared  with  sunlight,  259;  influ- 
ences on  the  earth,  262;  mass  of,  de- 
termined, 243,  244;  her  motion  (ap- 
parent), 228;  her  motion  relative  to 
the  sun,  241 ;  her  mountains,  measure- 
ment of  their  elevation,  270;  her  orbit 
with  reference  to  the  earth,  238;  her 
parallax  determined,  239;  perturba- 
tions, 448-461;  her  rotation  and  libra- 
tions,  248-251;  her  phases,  253;  her 
surface  character,  263-270;  culmina- 
tions for  longitude,  120,  A;  photo- 
graphs, 272;  heat  during  an  eclipse, 
377 ;  time  of  rising  or  setting,  131. 

Month,  the  anomalistic,  397,  note ;  the 
nodical,  397,  note ;  the  sidereal,  229, 
232;  the  synodic,  229,  232;  length  of, 
increased  by  perturbation,  453;  slight- 
ly shortened  by  the  secular  accelera- 
tion, 459. 

Motion,  direct  and  retrograde,  of  the 
planets,  '494;  of  the  solar  system  in 
space,  804-807 ;  in  line  of  sight,  effect 
on  spectrum,  321 ;  of  stars  in  line  of 
sight,  spectroscopically  observed,  802. 

Motions,  proper,  of  the  stars,  800-803. 

Mountains,  lunar,  their  height,  270. 

Mountain  method  of  determining  the 
earth's  density,  164. 

Multiple  stars,  882. 

Mural  circle,  the,  70l 

If. 

Nadir,  the,  denned,  9;  point  of  meridian 
circle,  67. 

Names  of  the  constellations,  792 ;  of  Jup- 
iter's satellites,  621;  of  satellites  of 
Mars,  590;  of  the  planets,  487,  489; 
of  Saturn's  satellites,  643,  note ;  of  the 
satellites  of  Uranus,  650;  of  stars,  794. 


Neap  tide  defined,  463. 

Nebula,  the  great,  in  Andromeda,  886; 
the  annular,  in  Lyra,  888;  of  Orion, 
the,  886,  892,  893. 

Nebulae,  the,  886-897;  changes  in,  892; 
their  distance,  896;  Lockyer's  meteoric 
theory,  894;  their  nature,  894;  their 
number  and  distribution,  895;  photo- 
graphs of ,  893;  planetary,  888;  spiral, 
888;  their  spectra,  and  chemical  ele- 
ments in  them,  890, 891. 

Nebular  hypothesis,  the,  908-915;  modi- 
fications of  the  original  theory,  912, 
913. 

Negative  eye-pieces  for  the  telescope,  44; 
shadow  of  the  moon,  381 ;  star  magni- 
tudes, 821. 

Neptune,  the  planet,  653-661 ;  anomalous 
retrograde  rotation,  in  relation  to  the 
nebular  hypothesis,  914;  appearance 
of  sun  and  solar  system  from  it,  658; 
(actual)  discovery  by  Galle,  654;  theo- 
retical discovery  by  Leverrier  and 
Adams,  653,  654;  its  discovery  "no 
accident, "  655 ;  the  computed  elements 
erroneous,  655;  its  satellite,  661 ;  spec- 
trum of,  660. 

NEWCOMB,  S.,  conclusions  as  to  sun's  age 
and  duration,  358, 359 ;  observations  on 
meteors,  776;  on  the  moon's  secular 
acceleration,  461;  on  the  structure  of 
the  heavens,  900;  his  value  of  the  so- 
lar parallax,  667;  velocity  of  light, 
225,  note,  668,  690. 

NEWTON,  Prof.  H.  A.,  daily  number  of 
meteors,  771 ;  investigation  of  meteoric 
orbits,  767,  785;  theory  of  the  consti- 
tution of  a  comet,  737. 

NEWTON,  SIR  ISAAC,  discovery  of  gravi- 
tation, 161,  419;  verification  of  the 
idea  of  gravitation  by  means  of  the 
moon's  motion,  419,  420;  discovery 
that  planetary  orbits  must  be  conies, 
421;  computation  of  a  cometary  orbit, 
700;  his  reflecting  telescope,  48. 

Nitrogen,  suspected  in  the  nebulae,  890. 

Node  of  an  orbit  defined,  233,  506. 

Nodes  of  moon's  orbit,  their  regression, 
455;  of  the  planetary  orbits,  their  mo- 
tion,  527. 

Nodical  month,  the,  249,  397,  note. 

NORDENSKIOI/D,  meteoric  ashes,  775. 

Nucleus  of  a  comet,  713,  716. 

Number  of  eclipses  in  a  year,  391;  in  a 
saros,  398 ;  of  meteors  and  meteorites, 
759,  769,  771 ;  and  designation  of  vari- 
able stars,  852. 

Nutation  of  the  earth's  axis,  214,  215. 


622 


INDEX. 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.] 


O. 

Oberon,  the  outer  satellite  of  Uranus,  650. 

Object-glass,  achromatic,  41;  residual  or 
secondary  spectrum  of,  42;  designed 
for  photography,  42. 

Oblateness,  or  ellipticity  of  a  spheroid,  150. 

Oblique  sphere,  the,  33. 

Obliquity  of  the  ecliptic  defined  and 
measured,  176;  of  the  ecliptic,  secular 
change  of,  197. 

Occult  ation,  circle  of  perpetual,  33;  of 
stars,  399  ;  of  stars  used  for  longitude 
determination,  120  C;  of  stars  proving 
absence  of  lunar  atmosphere,  256. 

OLBERS,  discovers  Pallas  and  Vesta,  593. 

OLMSTED,  D.,  his  researches  on  meteors, 
785. 

OPPOLZEB,  T.,  effect  of  meteors  on  the 
moon's  motion,  778  ;  orbit  of  TempePs 
Comet,  786;  motion  of  Winnecke's 
Comet,  711  ;  canon  of  eclipses,  390. 

Orbit  of  the  earth,  its  form  determined, 
182;  of  the  earth,  effect  of  meteors 
upon  it,  778;  of  the  earth,  perturba- 
tions, 197;  of  the  moon,  238;  of  the 
moon,  its  perturbations,  448-461  ;  of  a 
planet,  determined  graphically,  428, 
431,  432;  planetary,  its  elements,  505- 
510;  planetary,  its  elements,  determi- 
nation of,  519. 

Orbits  of  binary  stars,  873;  of  comets, 
700-709;  of  planets,  diagram,  489;  of 
sun  and  stars  in  the  stellar  system,  904. 

Origin  of  comets,  738-741;  of  meteorites 
or  aerolites,  767. 

Orthogonal  component  of  the  disturbing 
force,  445,  455. 


Pallas,  discovered  by  Olbers,  593. 

PALISA,  discoverer  of  sixty-five  asteroids, 
593. 

Parabola,  the,  described  as  a  conic,  422, 
423. 

Parabolic  comets,  their  number,  702;  ve- 
locity, the,  429. 

Parallax  (diurnal)  ,  defined  and  discussed, 
82,  83;  of  the  moon,  determined,  239; 
of  the  sun,  classification  of  methods, 
669;  of  the  sun,  gravitational  methods, 
687-389;  of  the  sun,  history  of  inves- 
tigations, 666-668;  of  the  sun,  method 
of  Aristarchus,  670;  of  the  sun,  meth- 
od of  Hipparchus,  671  ;  of  the  sun  by 
observations  on  Mars,  673,  676;  of  the 
sun  by  transits  of  Venus,  678,  686;  of 
the  sun  by  the  velocity  of  light,  690- 
692;  of  the  sun,  Ptolemy's  value, 


671;  of  the  stars  (annual),  808-814;  of 
a  Centauri,  Henderson,  809,  810;  of 
61  Cygni,  Bessel,  809-811;  of  a.  Lyra?, 
negative,  Pond,  809;  stellar,  absolute 
method,  810;  stellar,  differential  meth- 
od, 811;  stellar,  table  of,  Appendix, 
Table  IV. 

Parallactic  inequality  of  the  moon,  687: 
orbit  of  a  star,  808. 

Parallel  sphere,  the,  32. 

Parallels  of  declination,  23. 

PEIRCE,  B.,  heat  from  meteors,  355;  on 
the  mass  of  comets,  719;  theory  of 
sun  spots,  306. 

Pendulum,  compensation,  51;  use  in 
clocks,  50;  used  in  determining  form 
of  the  earth,  152-155;  free,  of  Foucault, 
showing  earth's  rotation,  139-141. 

Penumbra  of  the  earth's  shadow,  368; 
of  the  moon's  shadow,  383;  of  a  sun 
spot,  295. 

Perigee  and  apogee  defined,  238. 

Perihelia  of  comets,  their  distribution, 
706. 

Perihelion  of  earth's  orbit  defined,  183; 
its  motion,  199;  of  Mercury's  orbit,  its 
motion,  602. 

Period,  sidereal  and  synodic,  of  the  moon, 
229-232;  sidereal  and  synodic,  of  a 
planet,  defined,  490;  sidereal,  of  a 
planet,  determined,  513,  514. 

Periodic  comets,  703,  704,  738-740;  table 
of  comets  of  short  period,  Appendix, 
Table  III. 

Periodicity  of  sun  spots,  307-309. 

Persei,  P,  or  Algol,  848. 

Perseids,  the,  meteoric  swarm,  780,  782, 
783. 

Personal  equation,  114, 120  A,  121  B. 

Perturbations,  lunar,  448-461;  planetary, 
521-523;  of  Mars  and  Venus  by  the 
earth  as  determining  the  sun's  par- 
allax, 689. 

PETERS,  C.  H.  F.,  discovers  fifty-two 
asteroids,  593. 

PIAZZI,  discovery  of  Ceres,  592. 

PICARD,  measure  of  the  earth,  135,  419. 

PICKERING,  E.  C.,  his  meridian  photome- 
ter, 828;  photography  of  stellar  spec- 
tra, 860, 862 ;  photometric  observations 
of  the  eclipses  of  Jupiter's  satellites, 
630;  the  Harvard  photometry,  827. 

Phases  of  Mercury,  Venus,  and  Mars, 
559,  567,  582;  of  the  moon,  253. 

Phobos,  the  inner  satellite  of  Mars,  589. 

Photographs  of  the  moon,  272;  of  the 
nebulae,  893;  of  the  solar  corona,  328; 
of  the  sun's  surface  and  spots,  289. 


INDEX. 


623 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.'] 


Photographic  object-glasses,  42;  obser- 
vations of  eclipses  of  Jupiter's  satel- 
lites, 630,  and  note;  observations  of 
transit  of  Venus,  684-686. 

Photography  as  a  means  of  photometry, 
829 ;  solar,  289 ;  spectroscopic,  motion 
in  line  of  sight,  802 ;  applied  to  star- 
charting,  798 ;  in  determination  of  stel- 
lar parallax,  812;  of  stellar  spectra, 
859-863 ;  lunar,  272. 

Photometer,  the  meridian,  828;  polariza- 
tion, 827 ;  the  wedge,  826. 

Photometry,  Harvard,  the,  827,  828;  by 
means  of  photography,  829;  by  the 
spectroscope,  831 ;  of  sunlight,  332- 
335;  stellar,  823-831. 

Photosphere  of  the  sun,  its  nature,  291, 
292,  361. 

Photo-tachymetrical  determination  of 
the  sun's  parallax,  690-692. 

Physical  characteristics  of  comets,  712; 
method  of  determining  sun's  parallax, 
690-692. 

Planet,  intra-Mercurial,  602-606;  trans- 
Neptunian,  662. 

Planets  attending  certain  stars,  880,  881 ; 
distances  and  periods,  439;  enumer- 
ated, 486,  487 ;  relative  age,  according 
to  nebular  hypothesis,  913, 915 ;  orbits, 
diagram  of,  489;  orbits,  elements  of, 
505,  510 ;  motions,  apparent,  494-9. 

Planetoid.    See  Asteroid,  591. 

Planetary  data,  tables  of,  Appendix, 
Table  I. ;  data,  accuracy  of,  663;  neb- 
ulae, 888 ;  system,  facts  suggesting  the 
theory  of  its  origin,  907 ;  system,  Sir 
J.  Herschel's  illustration  of  its  dimen- 
sions, 664. 

Pleiades,  the,  884. 

POGSON,  the  absolute  scale  of  star-magni- 
tudes, 819. 

Pole  of  the  earth,  28;  (celestial),  defined, 
14;  its  altitude  equal  to  the  latitude, 
30,  100;  its  place  affected  by  preces- 
sion, 206,  207. 

Pole-star,  ancient,  a  Draconis,  207;  its 
position  and  recognition,  15. 

Polar  distance,  defined,  23;  point  of  me- 
ridian circle,  66. 

Position-angle  of  a  double  star,  868. 

Position  of  a  heavenly  body,  how  deter- 
mined, 128,  129. 

Positive  eye-pieces  for  telescopes,  44. 

POUILLET,  his  pyrheliometer,  340. 

Power,  magnifying,  of  telescope,  37. 

POYNTING,  determination  of  the  earth's 
density,  170. 

Practical  astronomy  defined,  2. 


Precession  of  the  equinoxes,  205-212. 

Prime  vertical  defined,  19;  vertical  in- 
strument, 62,  106. 

Priming  and  lagging  of  the  tides,  470. 

PRITCHARD,  PROF.  C.,  determination  of 
stellar  parallax  by  means  of  photog- 
raphy, 812;  stellar  photometry,  826; 
Uranometria  Oxoniensis,  826. 

PRITCHETT,  C.  W.,  discovery  of  the  great 
red  spot  on  Jupiter,  618. 

Problem  of  three  bodies,  437-461 ;  of  two 
bodies,  424-433. 

Problems  illustrating  Kepler's  third  law, 
413. 

PROCTOR,  R.  A.,  on  the  origin  of  comets, 
741 ;  determination  of  the  rotation 
period  of  Mars,  584. 

Projectiles,  deviation  caused  by  earth's 
rotation,  143 ;  their  path  near  the  earth, 
435. 

Projectile  force,  careless  use  of  the  term, 
401. 

Prominences,  or  protuberances,  the  so- 
lar, 291,  323-326,  363;  quiescent  and 
eruptive,  325,  326. 

Proper  motions  of  the  stars,  800,  803. 

Proximity  of  a  star,  indications  of  it,  813. 

Ptolemaic  system,  the,  500,  502. 

PTOLEMY,  his  almagest,  500,  700,  795. 

Pyrheliometer  of  Pouillet,  340. 


Quantity  of  the  solar  radiation  in  calo- 
ries, 338-340;  of  sunlight  in  candle 
power,  332,  333. 

Quiescent  prominences,  325. 

R. 

Radial  component  of  the  disturbing 
force,  446 ;  velocity,  802,  802*. 

Radian,  the,  defined  as  angular  unit,  5, 
note. 

Radiant,  the,  in  meteoric  showers,  780. 

Radius  of  curvature  of  a  meridian,  149. 

RANYARD,  peculiar  theory  of  the  repul- 
sive force  operative  in  comets'  tails, 
733. 

Rate  of  a  clock  defined,  53. 

Reading  microscope,  the,  64. 

Recognition  of  elliptic  comets,  difficul- 
ties, 704. 

Red  spot  of  Jupiter,  the,  618. 

Reduction  of  mean  star  places  to  appar- 
ent and  vice  versa,  797. 

Reflecting  telescope,  various  forms,  47, 48  ; 
telescopes,  large  instruments,  48. 


624 


INDEX. 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.] 


Refraction,  atmospheric,  its  law,  89,  90; 
determination  of  its  amount,  94,  95; 
effect  of  temperature  and  barometric 
pressure,  91 ;  effect  upon  form  and  size 
of  discs  of  sun  and  moon  near  the  hori- 
zon, 93;  effect  upon  time  of  sunrise 
and  sunset,  92. 

Refracting  telescope  (simple),  36;  tele- 
scope, achromatic,  41. 

Refractors  and  reflectors  compared,  49. 

REICH,  determination  of  the  density  of 
the  earth,  1GG;  experiments  upon  fall- 
ing bodies,  138. 

Relative  motion,  law  of,  492;  sizes  of  the 
planets,  diagram,  550. 

Repulsive  force  acting  on  comets,  728, 
731,  732,  734. 

Retardation  of  earth's  rotation  by  the 
tides,  4G1,  483. 

Reticle  used  in  telescope  for  pointing,  46. 

Retrograde  revolution  of  the  satellites 
of  Uranus  and  Neptune,  652,  661,  914. 

Reversing  layer  of  the  solar  atmosphere, 
291,  319,  320,  362. 

Reversal  of  the  spectrum,  314. 

Rhea,  the  fifth  satellite  of  Saturn,  643, 
note. 

Rigidity  of  the  earth,  171,  482. 

Right  ascension  defined,  25, 27;  ascension 
determined  by  transit  instrument,  59, 
128,  129 ;  sphere,  the,  31. 

Rings  of  Saturn,  637-642. 

ROSSE,  LORD,  observations  of  lunar  heat, 
260,  261,  377;  his  great  telescope,  48; 
spiral  nebulae,  888. 

Rotation,  distinguished  from  revolution, 
248,  248*;  of  the  earth,  affected  by  the 
tides,  461,  483;  of  the  earth,  proofs  of, 
138-143;  of  planets,  how  determined, 
543;  period  of  Jupiter,  615;  period  of 
Mars,  584;  period  of  the  moon,  248, 252; 
period  of  Saturn,  635;  of  the  sun,  281, 
283;  period  of  Venus,  570;  periods,  see 
also  Appendix,  Table  I. 


Saros,  the,  395-398;  number  of  eclipses  in 
a  saros,  398. 

Satellites  of  Jupiter,  621-631;  of  Mars, 
590,  591;  of  Neptune,  661;  of  Saturn, 
643,  644;  of  Uranus,  650-652;  general 
table  of,  Appendix,  Table  II. 

Satellite  orbits,  generally  circular,  548. 

Saturn,  the  planet,  632-644. 

Schehallien,  determination  of  the  earth's 
mass,  164. 


SCHIAPABELLI,  connection  between  com- 
ets and  meteors,  786;  his  map  cf  Mars, 
588. 

SCHROTER,  the  rotation  of  Mercury,  559; 
the  rotation  of  Venus,  570. 

SCHWABE,  discovery  of  the  periodicity  of 
sun  spots,  307. 

Scintillation  of  the  stars,  864. 

Sea,  ship's  place  at,  103, 120  B,  121  A,  124- 
126. 

Seasons,  the,  explained,  190,  192,  193; 
difference  between  northern  and 
southern  hemispheres,  194,  195. 

SECCHI,  theories  of  sun  spots,  303,  305; 
observations  on  stellar  spectra,  856, 
857. 

SEIDEL,  his  photometer,  827. 

Secular  acceleration  of  the  moon's  mean 
motion,  459-461;  changes  in  the  earth's 
orbit,  196-200;  perturbations  in  the 
planetary  system,  525-529. 

Semi-diameter,  augmentation  of  the 
moon's,  88;  correction  for,  in  sextant 
observations,  88. 

Semi-major  axis  of  a  planet's  orbit,  de- 
fined and  discussed,  505,  506;  axis  of 
the  planets'  orbits,  invariable,  526; 
axis  as  depending  on  planet's  velocity, 
428-430. 

Separating  power  of  a  telescope,  Dawes, 
43. 

Sequences,  method  of,  in  stellar  photom- 
etry, 824. 

Sextant,  the,  described,  76;  the,  used  to 
determine  latitude,  103;  the,  used  in 
finding  a  ship's  place  at  sea,  103,  116, 
125,  126;  the,  used  in  determining 
time,  115, 116. 

Shadow  of  the  earth,  its  dimensions,  367; 
of  the  moon,  379, 380;  of  the  moon,  its 
velocity  over  the  earth,  384. 

Ship's  place  at  sea,  determination  of,  103, 
120  B,  121  A,  124-126. 

Shooting  stars,  770,  787 ;  ashes  of,  775 ; 
brightness  of,  773 ;  comparative  num- 
bers in  morning  and  evening,  772 ;  daily 
number  of,  771 ;  elevation  of,  774 ;  mass 
of,  776 ;  materials  of,  775 ;  path  of,  774 ; 
showers  of,  780-786 ;  spectra  of,  775 ; 
velocity  of,  774. 

Short-period  comets,  703 ;  comets,  table 
of,  Appendix,  Table  III. 

Showers,  meteoric,  780-786. 

Sidereal  day  defined,  26, 110 ;  month,  229 ; 
time,  26,  110 ;  year,  its  length,  216. 

Siderostat,  Add.  A,  page  580  c. 

Signals  used  in  determining  difference  of 
longitude,  119. 


INDEX. 


625 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages,] 


Signs  of  the  zodiac,  177. 

Single-altitude  method  of  determining 
local  time,  116. 

Sirius  and  its  companion,  875;  its  light 
compared  with  the  sun's,  334,  832,835; 
its  mass,  877. 

Sky,  apparent  distance  of,  6. 

Slitless  spectroscope,  the,  860-862. 

Solar  constant,  the,  338-340;  eclipses,  319, 
323, 327,  329, 387-390,  393, 398;  eclipses, 
their  rarity,  398;  engine  of  Ericsson 
and  Mouchot,  345 ;  eye-pieces,  286, 287; 
parallax,  see  Parallax  of  the  Sun; 
system,  age  of,  922:  time,  apparent 
and  mean,  defined,  111,  112. 

Solstice  defined,  176. 

SOSIGENES,  devises  the  Julian  calendar, 
219. 

Spectra  of  comets,  724-726;  of  meteors, 
775;  of  nebulae,  890, 891;  of  stars,  855- 
863. 

Spectroscope,  principles  of  its  construc- 
tion, 311-313 ;  how  it  shows  the  solar 
prominences,  324 ;  slitless,  860-862. 

Spectroscopic  measurement  of  motions  in 
the  line  of  vision,  321,  802,  879,  879*. 

Spectrum  of  the  chromosphere  and  prom- 
inences, 323 ;  of  the  corona,  329 ;  solar 
(photosphere),  312;  solar,  compared 
with  iron,  315;  of  sun  spots,  321; 
analysis,  fundamental  principles,  314; 
photometry,  831. 

Sphere,  the  celestial,  conceptions  of  it,  4 ; 
the  oblique,  33;  the  parallel,  32;  the 
right,  31. 

Spheres,  attraction  of,  162. 

Spheroid,  terrestrial,  its  dimensions,  145, 
Appendix,  page  527. 

Spherical  aberration  of  a  lens,  39 ;  as- 
tronomy, defined,  3;  shell,  its  attrac- 
tion, 169. 

Spider  lines  in  a  reticle,  46. 

Spring  tide  defined,  etc.,  463. 

Spurious  disc  of  stars  in  the  telescope,  43. 

Stability  of  the  planetary  system,  530- 
533. 

Standard  and  local  time,  122. 

Stars,  binary,  see  Binary  Stars ;  causes 
of  the  difference  in  their  brightness, 
836 ;  colors  of,  830 ;  dark,  836 ;  desig- 
nations and  names,  794;  their  real 
diameters,  837;  distribution  of,  899; 
double,  see  Double  Stars;  gradual 
changes  in  their  light,  839 ;  heat  from 
them,  834;  light  compared  with  sun- 
light, 334,  832,  835;  magnitudes  of, 
816-822 ;  missing,  840 ;  nature,  as  be- 
ing suns,  789;  number  of,  790;  par- 


allax and  distance,  808-814;  photog- 
raphy of,  798;  photometric  observa- 
tions of,  823-831 ;  proper  motions  of, 
800-803 ;  proximity  of,  its  indications, 
813;  seen  by  day  with  telescope,  38; 
shooting,  see  Shooting  Stars ;  tempo- 
rary, 842-845;  triple  and  multiple, 
882 ;  twinkling  of,  or  scintillation,  864 ; 
variable,  see  Variable  Stars. 

Star-atlases,  793. 

Star-catalogues,  795. 

Star-charts,  798. 

Star-clusters,  883,  884. 

Star -gauges,  899. 

Star-motions,  799-803. 

Star-places,  how  affected  by  aberration, 
etc.,  226;  their  determination,  796; 
mean  and  apparent,  797. 

Statical  theory  of  the  tides,  469. 

Station  errors,  158. 

Stellar  spectra,  855,  856;  classification 
of,  857,858;  photography  of,  859-863; 
system,  the  hypothetical,  901-904. 

STONE,  E.  J.,  attempted  observation  of 
stellar  heat,  834. 

Stripe,  central,  in  comets'  tails,  730. 

Structure  of  the  heavens,  900-904. 

STBUVE,  VON,  F.  G.  W.,  on  distribution  of 
stars,  899. 

STBUVE,  VON,  LTJDWIG,  investigation  of 
sun's  motion  in  space,  806. 

STBUVE,  VON,  OTTO,  Saturn's  rings,  637, 
642. 

SUMNEB,  CAPT.,  his  method  of  determin- 
ing a  ship's  place  at  sea,  125,  126. 

Sun,  the,  273-364 ;  age  and  duration  of, 
359,  922 ;  apparent  annual  motion  of, 
172, 173;  attraction  on  the  earth,  its 
intensity,  436;  candle  power  of  sun- 
light, 332,  333 ;  chemical  elements  in 
it,  315-317;  diameter,  surface,  and 
volume,  276,  277;  distance  and  par- 
allax, 274,  275,  665-693 ;  gravity  at  its 
surface,  280 ;  heat  emission,  338-357 ; 
light,  332-336;  mass  and  density, 
278,  279 ;  its  motion  in  space,  804-806 ; 
physical  constitution,  360-364 ;  its  tem- 
perature, 349-351 ;  the  central,  807. 

Sun  spots,  their  development  and  chang- 
es, 297,  298;  distribution  on  sun's  sur- 
face, 301;  general  description,  293, 
294;  influence  on  terrestrial  condi- 
tions, 309,310;  periodicity  of,  307-309; 
their  spectrum,  321;  theories  as  to 
their  cause  and  nature,  302-306. 

Sun's  way,  apex  of,  804-806 

Sunrise  and  sunset  affected  by  refrac- 
tion, 92. 


626 


INDEX. 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.] 


Superior  planet,  motions  of,  497. 
Surface  errors  in  lenses  and  mirrors,  49  ; 

of  the  moon,  263-270. 
Swarms,  meteoric,  783. 
SWEDENBORG,  a  proposer  of  the  nebular 

hypothesis,  908. 
System,  planetary,  facts    suggesting   a 

theory  of   its   origin,  907;    numerical 

data,  Appendix,  Table  I.  ;  stellar,  901- 

904. 
Synodic  month,  or  revolution,  of  moon, 

229-231;  period,  general  definition  of, 

490. 
Syzygy,  defined,  230. 


Tables,  Appendix.  Greek  alphabet,  page 
601  ;  miscellaneous  symbols,  page  601  ; 
dimensions  of  the  earth,  page  601; 
time  constants,  page  602;  Table  I., 
elements  of  solar  system,  page  603; 
Table  II.,  satellites  of  the  system, 
pages  604,  605  ;  Table  III.,  short-period 
comets,  page  606;  Table  IV.,  par- 
allaxes of  stars,  page  607  ;  Table  V., 
orbits  of  binary  stars,  page  608  ;  Table 
VI.  ,  the  variable  stars,  page  609  ;  Table 
VII.,  radial  motions  of  stars,  page  610. 

Tables,  in  body  of  the  book.  The  constel- 
lations, 792;  approximate  distances 
and  periods  of  the  planets,  489;  dis- 
tance of  sun  corresponding  to  certain 
values  of  the  parallax,  668  ;  distribu- 
tion of  stars  with  reference  to  the 
galaxy,  899  ;  iron  meteors  seen  to  fall, 
758  ;  naked-eye  stars  north  of  celestial 
equator,  818  ;  orbits  and  masses  of  cer- 
tain binary  stars,  877;  parallaxes  of 
first-magnitude  stars,  Elkin,  815  ;  prop- 
er motions  of  certain  stars,  800  ;  signs 
of  the  zodiac,  177  ;  telescopic  aperture 
required  to  show  stars  of  given  mag- 
nitude, 822  ;  temporary  stars,  842  ;  to- 
tal light  from  stars  of  different  mag- 
nitude, 833  ;  velocity  of  free  wave  at 
various  depths,  473. 

Tails  or  trains  of  comets,  713,  717,  728- 
736. 

TALCOTT,  CAPT.,  his  zenith  telescope,  105. 

Tangential  component  of  the  disturbing 
force,  447. 

Telegraph  used  in  determination  of  lon- 
gitude, 121  B. 

Telescope,  the,  achromatic,  41;  distinct- 
ness of  image,  39  ;  equatorial,  72  ;  eye- 
pieces, 44  ;  the  general  theory,  35  ;  in- 
vention of,  35  ;  light-gathering  power, 


38;  long,  of  Huyghens,  40;  magnify- 
ing power,  37;  object-glass,  various 
forms,  41, 42;  reflecting,  various  forms, 
47,  48;  refracting,  simple,  36;  relation 
of  its  aperture  to  the  "magnitude" 
of  the  smallest  star  visible  with  it, 
822 ;  separating  or  dividing  power,  43. 

Telespectroscope,  313. 

Temperature,  cause  of  the  annual  change, 
192,  193;  of  the  moon,  261 ;  of  the  sun, 
349-351. 

Temporary  stars,  842-845,  580  e. 

"Terminator,"  the,  on  the  moon's  sur- 
face, its  form,  253. 

Tethys,  the  third  satellite  of  Saturn,  643, 
note. 

THOMSEN  of  Copenhagen,  the  mechanical 
equivalent  of  light,  776. 

THOMSON,  SIR  W.,  on  the  temperature  of 
meteors,  765 ;  rigidity  of  the  earth,  171, 
482. 

Three  bodies,  the  problem  of,  437,  438. 

Tidal  evolution,  484,  916 ;  friction,  effect 
on  the  earth's  rotation,  461,  483; 
wave,  its  origin  and  course,  476. 

Tides,  the,  definition  of  terms  relating  to 
them,  463;  priming  and  lagging  of,  470; 
statical  theory  of,  469;  wave  theory 
of,  472. 

Tide-raising  force,  the,  464-467. 

Time,  defined  as  an  hour-angle,  109;  its 
determination  by  the  sextant,  115, 116; 
its  determination  by  the  transit  in- 
strument, 113;  equation  of,  explained 
and  discussed,  201-204;  sidereal,  de- 
fined, 26,  110;  solar,  apparent,  111; 
solar,  mean,  112;  standard  and  local, 
122. 

TISSERAND  on  peculiarities  of  satellite 
orbits,  548. 

Titan,  the  sixth  and  great  satellite  of 
Saturn,  643. 

Titania,  the  third  satellite  of  Uranus,  650. 

TODD,  PROF.  D.  P.,  search  for  trans-Nep- 
tunian planet,  662. 

Torsion  balance,  determination  of  the 
earth's  density,  165. 

Trade  winds,  proving  rotation  of  the 
earth,  143. 

Trains  of  meteors,  766,  773. 

Transits  of  moon  across  meridian,  the 
interval  between  them,  235;  of  Mer- 
cury, 561,  562 ;  of  Venus,  law  of  recur- 
rence, 575-577;  of  Venus,  xiseti  for  de« 
termination  of  solar  parallax,  678-686. 

Transit  circle,  see  Meridian  Circle,  63; 
instrument,  59-61;  instrument  used 
in  determining  time,  113. 


INDEX. 


627 


[All  references,  unless  expressly  stated  to  the  contrary,  are  to  articles,  and  not  to  pages.] 


Trans-Neptunian  planet,  hypothetical, 
662. 

Transparency  of  space  as  affected  by 
meteors,  779. 

Triple  and  multiple  stars,  882. 

Tropics,  defined,  176. 

Tropical  year,  its  definition  and  its 
length,  216. 

Twilight,  theory  and  duration  of,  96,  97, 
130. 

Twinkling,  or  scintillation,  of  the  stars, 
864. 

Two  bodies,  problem  of,  424-433. 

TYCHO  BRAKE  discovers  the  lunar  varia- 
tion,457  ;  observations  of  comet  of  1577, 
700 ;  temporary  star  in  Cassiopeia,  843 ; 
his  planetary  system,  504. 

U. 

Umbriel,  the  second  satellite  of  Uranus, 

650. 
Unit  of  stellar  distances,  the  light-year, 

814. 

TJranolith.    See  Meteorite. 
Uranometria    Nova:    Argelander,    817; 

Oxoniensis,  826. 
Uranus  and  Neptune,  their  anomalous 

rotation   in  relation  to  the  nebular 

hypothesis,  914 ;  the  planet,  645-352. 
Utility  of  astronomy,  2. 

V. 

VAN  DER  KOLK'S  theorem,  434. 

Vanishing  point  of  a  system- of  parallel 
lines,  7. 

Variable  nebulae,  889. 

Variable  stars,  838-854 ;  classification  of, 
838;  explanation  of  their  variation, 
849-851 ;  methods  of  observation,  854 ; 
their  number  and  designation,  852; 
their  range  of  variation,  853. 

Variation,  the  lunar,  457. 

Vega,  or  a  Lyrse,  its  light  compared  with 
the  sun's,  334,  832,  835 ;  its  spectrum, 
859. 

Velocity  of  air  currents  at  high  elevations, 
773,  note;  areal,  linear  and  angular, 
law  of,  407-409;  of  earth  in  her  orbit, 
225,  note,  278;  of  light,  225,  note, 
690-692;  of  the  moon's  shadow,  384; 
parabolic,  or  velocity  from  infinity, 
429 ;  of  planet  at  any  point  in  its  orbit, 
434;  of  stellar  motions,  801. 

Venus,  the  planet,  563-577;  her  atmos- 
phere and  its  effect  upon  observations 
of  a  transit,  681 ;  transits  of,  used  to 
determine  solar  parallax,  678-686. 


Vertical,  angle  of  the,  156;  circles  de- 
fined, 12. 

VOGEL,  his  classification  of  stellar  spec- 
tra, 858 ;  star  motions  in  line  of  sight, 
802,  863. 

Vulcan,  hypothetical  intra-Mercurial 
planet,  603,  604. 

W. 

Waste  of  solar  energy,  347. 

Water,  absence   of,  on  the  moon,  258; 

presence  of,  in  atmosphere  of  planets, 

560,  573,  589. 
WATSON,  J.  C.,  discovers  and  endows 

twenty- two  asteroids,  593,  601. 
Wave-length  of  a  ray  of  light  affected  by 

motion  in   the  line  of  vision  —  Dop- 

pler's  principle,  321,  note. 
Wave-theory  of  the  tides,  472. 
Weather,  moon's  influence  on  it,  262. 
Wedge  photometer,  the,  826. 
Weight,  loss   of,  between  equator  and 

pole,  152-155;    and  mass,  distinction 

between  them,  159,  160. 
WILSING,  determination  of  the  earth's 

density,  167. 

WINNECKE'S  comet,  acceleration  of,  711. 
WOLF,  periodicity  of  the  sun  spots,  307. 
WORMS,  formula  for  the  eastward  devia- 
tion of  a  falling  body,  138. 

Y. 

Year,  bissextile,  or  leap,  219 ;  beginning 
of,  222 ;  of  confusion,  219 ;  eclipse,  391 ; 
sidereal,  tropical,  and  anomalistic,  216, 
also  Appendix,  page  602 ;  of  light,  unit 
of  stellar  distance,  814;  fictitious,  223. 

z. 

Zenith,  the  astronomical  and  geocentric, 
8. 

Zenith  distance  defined,  21;  telescope, 
for  determination  of  latitude,  105. 

ZENKER,  theory  of  a  comet's  constitu- 
tion, 733. 

Zero  points  of  a  meridian  circle,  66,  67. 

Zodiac,  the,  and  its  signs,  177 ;  signs  of, 
as  affected  by  precession,  208. 

Zodiacal  light,  the,  607,  608. 

ZOLLNER,  albedo  ofc  the  planets,  558, 
572,  583,  614,  636,  648,  660,  also  Appen- 
dix, page  529 ;  his  photometer,  827 ;  on 
the  repulsive  force  acting  upon  comets, 
732. 


SUPPLEMENTARY    INDEX. 


(Unless  otherwise  specified  all  references  are  to  Articles.') 


ALBRECHT,  diagram  of  the  variation  of 

latitude,  108. 
Algol,  diameter  of,  837 ;  light-curve,  848 ; 

system  of,  851. 

Almucantar,  Dr.  Chandler's  equal-alti- 
tude instrument,  105. 
Arequipa,  observations  at,  798*,  854*. 
Asteroids,  diameters  of,  596;  mass  of, 

597-8;    observed  for  solar  parallax, 

676*. 
Atmosphere,  density  depending  on  mass 

of  heavenly  body,  272*. 
Azimuthal  motion  of  stars  at  the  horizon, 

1001. 

BAILEY,  S.  I.,  discovery  of  variable  star- 
clusters,  854*. 

BALMER,  law  of  wave-lengths  in  the 
spectrum  of  hydrogen,  861,  865*. 

BARNARD,  E.  E.,  discoveries  of  comets, 
698,  740*,  752* ;  discovery  of  the  fifth 
satellite  of  Jupiter,  621 ;  measures  of 
the  diameter  of  asteroids,  596 ;  obser- 
vation of  the  eclipse  of  lapetus,  641*. 

BELOPOLSKY,  spectroscopic  observations, 
802,  802*,  851. 

Binaries,  spectroscopic,  879,  879*;  tidal 
evolution  of,  877. 

Bolograph,  343. 

BOYS,  C.  V.,  determination  of  the  "  New- 
tonian Constant"  or  Constant  of  Grav- 
itation, 161;  on  the  density  of  the 
earth,  166. 

BROOKS,  W.  R.,  comet  discoveries,  697-8. 

Bruce  photographic  telescope,  the,  798*, 
861. 

Calculation  of  a  lunar  eclipse,  1005. 

CAMPBELL,  W.  W.,  observations  on  the 
spectrum  of  Mars,  589. 

Candle,  decimal,  333. 

Canon  of  eclipses,  Th.  Oppolzer,  390. 

Cape  of  Good  Hope,  observations  for 
solar  parallax,  676*;  great  photo- 
graphic telescope,  798*. 


CHANDLER,  S.  C.,  researches  on  comet 
1889  V,  740* ;  on  variation  of  latitude, 
108;  on  the  system  of  Algol,  851. 

CHARLOIS,  photographic  discovery  of 
asteroids,  594. 

Ccelostat,  Add.  A.,  p.  580 d. 

Comet,  Holmes's,  714,  723,  726;  Lexell- 
Brooks,  740* ;  Pons-Brooks,  697 ;  Tern- 
pel's,  739,  786-7 ;  Turtle's,  786. 

Comets,  families  of,  739 ;  "  home  of  the/5 
741* ;  photography  of,  752*. 

COMMON,  A.  A.,  his  five-foot  reflector,  48 ; 
his  photographs  of  nebulae,  886. 

Conies,  proof  that  the  orbits  of  bodies 
moving  under  gravitation  are,  1006. 

Constant  of  Gravitation,  161, 166. 

Coronium,  unidentified  element  in  the 
solar  corona,  329. 

CREW,  H.  C.,  spectroscopic  observations 
on  the  rotation  of  the  sun,  283. 

Decimal  candle,  333. 

DENNING,    E.    J.,    stationary   meteoric 

radiants,  787*. 
DESLANDRES,  H.,  photography  of  solar 

prominences,  326*. 
DUNER,  N.  C.,  spectroscopic  observations 

on  the  rotation  of  the  sun,  283. 

Eclipse,  lunar,  projection  and  calculation 

of,  1004-5. 

Eclipse  of  1898,  page  267,  note. 
Eclipses,  canon  of,  Th.  Oppolzer,  390. 
ELKIN,  W.  L.,  asteroid  observations  for 

solar  parallax,  676*. 
Epicycloidal  relative  orbits  of  planets, 

1009. 

Equatorial  acceleration  of  sun,  310*. 
Eros,  note,  page  377. 

Fictitious  year,  the,  222. 

FIZEAU,  H.  L.,  Doppler-Fizeau  principle, 

321*. 
FLAMMARION,  C.,  speculations  respecting 

Mars,  589*,  589**. 


SUPPLEMENTARY   INDEX. 


629 


(Unless  otherwise  specified  all  references  are  to  Articles.) 


Flash  spectrum  at  an  eclipse,  319,  note; 

pag'e  "267,  note. 
FROST,  E.  B.,  thermal  radiation  of  sun 

spots,  301*. 

Gases  occluded  in  meteorites,  767. 
Gemination  of  the  canals  of  Mars,  588. 
Gravitation,  constant  of,  161, 166. 
GRAY,  WILSON  and,  measure  of  solar 
temperature,  351. 

H  and  K  lines,  in  spectrum  of  the 
chromosphere  and  prominences,  323, 
326* ;  of  the  corona,  329 ;  of  sun-spots, 
321 ;  in  stellar  spectra,  857,  note. 

HALE,  G.  E.,  solar  photography,  326*; 
spectroheliograph  work,  Add.  A. 

HARKNESS,W., solar  parallax  and  related 
constants,  155,  693,  note. 

Harvard  photography,  of  the  moon, 
272 ;  of  stars,  798* ;  of  stellar  spectra, 
859-861,  879-879*. 

HUMPHREYS  and  MOHLER,  effect  of  pres- 
sure in  shifting  spectrum-lines,  802*. 

Julian  Period,  Epoch,  and  Day,  223*. 
Jupiter,  discovery  of  the  fifth  satellite, 
621. 

KAYSER,  H.,  and  RUNGE,  C.,  series  of 

lines  in  spectra,  865*. 
KEELER,  J.E.,  spectroscopic  observations 

of  the  rotation  of  the  rings  of  Saturn, 

641* ;  of  the  radial  velocity  of  nebulae, 

891. 
Kepler's  Problem,  demonstration  and 

numerical  solution,  1002-3;  graphical 

solution,  1002*. 
KUSTNER,  F.,  discovery  of  the  variation 

of  latitude,  108. 

LA  PLACE,  P.  S.,  proposer  of  the  capture 
theory  of  comets,  740*. 

Latitude,  variation  of,  108. 

LOCKYER,  SIR  NORMAN,  spectroscopic 
observations  of  variable  stars,  851, 
879*. 

LOWELL,  P.,  measure  of  polar  compres- 
sion of  Mars,  586;  observations  and 
speculations  respecting  Mars,  588- 
589***;  observations  on  surface  and 
rotation  of  Mercury,  559 ;  on  surface 
and  rotation  of  Venus,  569,  570. 

Lick  Observatory,  telescope,  72 ;  photo- 
graphs of  the  moon,  272;  measures  of 
asteroids,  596 ;  discovery  of  the  fifth 
satellite  of  Jupiter,  621. 


Lunar  Eclipse,  projection  and  calcula- 
tion, 1004-5. 

Mass,  measured  by  its  inertia,  171*. 
Mercury,  motion  of  its  perihelion,  602; 

its  rotation,  559. 
Meteorites,  gases  in,  767. 
Meteoritic  constitution  of  Saturn's  ring, 

641*. 
Meudon,  great  photographic  telescope, 

798*. 
MICHELSON,  A.  A.,  micrometric  measures 

by  means  of  diffraction  fringes,  837. 
MINCHIN,  G.  M.,  use  of  selenium  in  stellar 

photometry,  831. 
MOHLER,    HUMPHREYS    and,  effect   of 

pressure  on  spectrum  lines,  802*. 
MouLTON,F.R.,planetesimal  theory  ,926*. 
MULLER,  G.,   and  KEMPF,    P.,  photo- 
metric catalogue,  827. 

NEWCOMB,  S.,  values  of  solar  parallax, 

693,  note. 

Newtonian  Constant,  the,  161. 
Nova  Persei,  Add.  B.,  580  e. 

Oases,  on  Mars,  588. 

Orbit,  of  body  moving  under  gravitation, 
proof  that  it  is  a  conic,  1006 ;  relative 
of  planet,  1009. 

OPPOLZER,  E.,  theory  of  sun-spots,  306. 

OPPOLZER,  Th.,  canon  of  eclipses,  390. 

Orion,  nebula  of,  photographs,  886 ;  spec- 
trum of,  890-1. 

Oxygen  in  the  Sun,  317,  note. 

Parabolic  Velocity,  defined,  429 ;  at  sur- 
face of  bodies,  272*,  435;  proof  of 
formula,  1008-9. 

Parameter,  of  an  orbit,  423,  1006. 

Parallax,  solar,  from  observations  of 
asteroids,  676*;  Newcomb's  table  of 
results, 693,  note  ;  stellar,  photographic 
determination,  812. 

PEIRCE,  B.,  the  "home  of  the  comets," 
741*. 

Photographic  telescopes,  Bruce,  798*, 
861 ;  Meudon,  798*. 

Photography,  applied  to  determination 
of  stellar  parallax,  812 ;  to  discovery 
of  asteroids,  594 ;  of  comets,  752* ;  to 
time-observations,  114. 

Photographs,  and  photographic  maps,  of 
the  moon,  272;  of  solar  prominences, 
326* ;  of  the  flash  spectrum,  319,  note. 

Photometry,  stellar,  Potsdam  photo- 
metric catalogue,  827;  use  of  sele- 
nium in,  831. 


630 


SUPPLEMENTARY   INDEX. 


(Unless  otherwise  specified  all  references  are  to  Articles.) 


Photo-visual  objective,  42*. 

PICKERING,  E.  C.,  discovery  of  spectro- 
scopic  binaries,  879,  879*;  series  in 
spectra,  865*. 

Planetesimal  theory,  926*. 

Pole  of  the  earth,  its  motion,  108. 

POOR,  C.  L.,  investigations  on  the  so- 
called  "  Lexell-Brooks  comet,"  740*. 

Potsdam,  Astro-physical  observatory, 
photometric  work,  827 ;  spectrograph, 
802*;  spectroscopic  work,  802*,  851, 
858,  879*. 

Pressure,  effect  on  wave-length,  802*. 

Projection  of  a  lunar  eclipse,  1004. 

Prominences,  solar,  photographs  of, 
326*;  white,  323,  note. 

Quartz  filaments,  46, 166. 

Radial  velocity  of  stars,  spectroscopic 
determination,  802,  802*,  Table  VII, 
page  610 ;  of  nebulae,  891. 

Radiants,  meteoric,  list  of  principal,  783 ; 
stationary,  787*. 

Refraction,  Table  of,  page  611. 

Rising  or  setting  of  a  heavenly  body, 
azimuthal  motion  at,  141, 1001 ;  com- 
putation of  the  moment,  130,  131. 

ROBERTS,  I.,  photographs  of  nebulas  and 
clusters,  884,  886. 

RUNGE,  C.,  line-series  in  spectrum  of 
helium,  865* ;  oxygen  in  the  sun,  317, 
note. 

RUTHERFURD,  L.  M.,  photographs  of  the 
moon,  272;  observation  of  stellar 
spectra,  856. 

SAMPSON,  R.  A.,  on  sun's  equatorial  ac- 
celeration, 310*. 

Satellite,  fifth,  of  Jupiter,  621. 

SCHAEBERLE,  J.  M.,  theory  of  sun-spots 
and  corona,  306. 

SCHIAPARELLI,  G.  V.,  canals  of  Mars, 
588;  rotation  of  Mercury,  559;  of 
Venus,  570. 

SEE,  T.  J.  J.,  evolution  of  binaries,  877 ; 
orbit  of  70  Ophiuchi,  882. 

Selenium  used  in  stellar  photometry,  831. 

Series  of  lines  in  spectra,  865*. 

SHACKLETON,  W.,  photograph  of  the  flash 
spectrum,  319,  note. 

Sidereal  time,  reduction  to  solar,  and 
the  converse,  1000. 

Sines,  curve  of,  used  in  graphical  solu- 
tion of  Kepler's  problem,  1002*. 

Spectroheliograph,  326*,  Add.  A.,  580  a. 


Spectroscopic  observations  on  the  sun's 
rotation,  283 ;  proof  of  constitution  of 
rings  of  Saturn,  641*;  of  flash  spec- 
trum, 319,  note ;  page  267,  note. 

SPOERER,  F.  W.  G.,  law  of  sun-spot  lati- 
tudes, 307*. 

Star,  time  of  rising  or  setting,  131. 

Stationary  meteoric  radiants,  787*. 

STONEY,  J.,  carbon  in  solar  photosphere, 
292,  note ;  limits  of  lunar  and  planet- 
ary atmospheres,  272*. 

STRUVE,  H.,  mass  of  Saturn's  rings,  641* ; 
polar  compression  of  Mars,  586. 

Sun-spots,  radiation  of,  301*. 

SWIFT,  L.,  discovery  of  comets,  698, 740*. 

TACCHINI,  P.,  white  prominences,  323, 
note. 

Thwartwise  velocity  of  stars,  801,  Table 
IV,  Appendix. 

Tidal  evolution  of  binary  systems,  877. 

Tide  caused  by  the  variation  of  latitude, 
108. 

Time,  sidereal  and  solar,  reduction  of 
each  to  the  other,  1000. 

TISSERAND,  F.  F.,  on  stationary  radiants, 
787* ;  on  system  of  Algol,  851. 

Two-bodies,  problem  of,  proof  that  the 
orbit  must  be  a  conic,  1006;  demon- 
stration for  the  velocity  of  a  body  at 
any  point  in  its  orbit,  1007. 

Variable-star  clusters,  854*.  Add.  C.,  5800 

Velocity,  molecular,  of  gases,  272* ;  para- 
bolic or  critical  at  surface  of  bodies, 
272*,  435,  1007;  orbital,  of  a  body 
moving  in  a  conic,  1008. 

Venus,  axial  rotation,  570 ;  surface  mark- 
ings, 569. 

VOGEL,  H.  C.,  system  of  Algol,  851;  of 
Spica,  classification  of  spectra,  859. 

Wave-length,  effect  of  pressure  upon, 
802*. 

Welsbach  burner,  illustrating  the  radi- 
ating action  of  the  solar  photosphere, 
361. 

WILSING,  J.,  on  sun's  equatorial  acceler- 
ation, 301*. 

WOLF,  M.,  photographic  discovery  of 
asteroids,  594. 

Wolf-Rayet  stars,  857. 

Yerkes  telescope,  49,  822. 


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Russell:  Glaciers  of  North  America 1.75 

Russell:  Lakes  of  North  America 1.50 

Trafton :   Laboratory  and  Field  Exercises  in  Physical  Geog- 
raphy     40 

Ward :   Practical  Exercises  in  Elementary  Meteorology      .     .  1.12 
Wright:   Field,  Laboratory,  and  Library  Manual  of  Physical 

Geography i.oo 

146  b 

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